Method for learning switching linear dynamic system models from data

ABSTRACT

From a set of possible switching states and responsive to a sequence of measurements, a corresponding sequence of switching states is determined for a system having a plurality of dynamic models, associates each model with a switching state such that a model is selected when its associated switching state is true. A state transition record is determined, based on the measurement sequence. The sequence of switching states is determined by backtracking through the state transition record. Alternatively, the switching state model is decoupled from the dynamic system model. The decoupled switching state model is transformed into a hidden Markov model (HMM) switching state model, while the decoupled dynamic system model is transformed into a time-varying dynamic system model. A solution to the dynamic system model is estimated using a Kalman filter. Next, variational parameters of the HMM switching state model are determined based on the estimated-solution, where the variational parameters measure an agreement of each model from the plurality of dynamic models with the solution. A sequence of switching states for the HMM switching state model is then determined based on the variational parameters of the HMM switching state model. finally, variational parameters of the dynamic system model are determined based on the determined sequence of switching states, such that the variational parameters are proportional to a combination of model parameters form the plurality of dynamic models weighted by the probability of the switching states.

RELATED APPLICATION(S)

This Application claims the benefit of U.S. Provisional Application No. 60/154,384, filed Sep. 16, 1999, the entire teachings of which are incorporated herein by reference.

BACKGROUND OF THE INVENTION

The problem of fitting dynamic models to sequences of data is a key technology in many applications. In econometrics and forecasting applications, data sequences may represent product inventory in a distribution channel over time, or the price of a stock or other financial instrument over time. In human motion applications, data sequences could represent the pose of the human body over time. For example, a motion such as a ballet plié can be described as a sequence of smooth changes in the angles of the arms and legs and pose of the torso.

Dynamic models can also be applied to modeling of spatial data sequences, such as genes or constrained kinematic chains.

The most basic example of learning the parameters of a dynamic model from data is the system identification problem for a single linear dynamic model. System identification is described in Ljung et al., “Theory and Practice of Recursive Estimation,” MIT Press, 1983. Unfortunately, a single linear model is incapable of representing a broad range of interesting data sequences.

A switching linear dynamic system (SLDS) model consists of a set of linear dynamic models and a switching variable that determines which model is in effect at any given point in time. A “fully connected” SLDS further assumes that there are temporal dependencies between the values of the switching variable at different times, as well as the states of different linear dynamic models. SLDS models are attractive because they can describe complex dynamics using simple linear models as building blocks. Given an SLDS model, the inference problem is to estimate the sequence of model states that best explains a measurement data sequence. Unfortunately, exact inference in SLDS models is computationally intractable, due to the large number of possible combinations of linear models over time.

Consider the case where there are r linear models in an SLDS model, and assume that the goal is to infer which model best explains each element in a measurement data sequence of length n. For the first element there are r possibilities. For two sequential elements there are r*r possibilities. There are r^(n) total possibilities for the entire data set or sequence of n elements. It is infeasible to examine each of these possibilities to determine the exact, optimal solution.

One prior method for inference using fully connected SLDS models is described in Bar-Shalom et al., “Estimation and Tracking: Principles, Techniques, and Software,” Artech House, Inc. 1993 and in Kim, “Dynamic Linear Models With Markov-Switching,” Journal of Econometrics, volume 60, pages 1-22, 1994. In this method, approximate inference is achieved by truncating or collapsing the number of discrete components in the evolving model. The models are used to detect different motion regimes while tracking a maneuvering target.

Additional approximate smoothing of the switching states is described in Kim. Neither of these two references tackle smoothing of the linear dynamic system states.

In most applications, it is not practical to build an SLDS model by hand and it is therefore desirable to learn the parameters of these models from training data. A method for SLDS learning is described in Shumway et al., “Dynamic Linear Models with Switching,” Journal of the American Statistical Association, 86(415), pages 763-769, September 1991. It assumes that the SLDS is not fully connected, i.e., the switching variable has no temporal dependencies. It also assumes that a prior distribution for the switching variable is known for each time instant. These assumptions do not hold for a broad range of practical applications.

Krishnamurthy et al., “Finite-dimensional Filters for Passive Tracking of Markov Jump Linear Systems,” Automatica, 34(6), pages 765-770, 1998, assumes that the switching variable follows a Markov process model and that observations of the switching variable are available for each time instant. However, for a broad range of practical applications, these observations are not available.

In another method, the switching variable determines which linear model is coupled to the measurement at each time instant. See Ghahramani et al., “Variational Learning for Switching State-Space Models,” which will appear in the journal Neural Computation. This method can produce decoupled linear models which reach steady-state before the data series is adequately modeled. It stands in contrast to methods with fully coupled SLDS in which all models are coupled through a single state space.

Pavlovic, Frey and Huang, “Time-series Classification Using Mixed-State Dynamic Bayesian Networks,” Proc. of Computeor Vision and Pattern Recognition, pages 609-615, June, 1999, considers a single linear dynamical model whose input is modeled as a discrete Markov process. This model explains all measurement variability as a consequence of the changes in input, which may not be true in general.

Blake et al., “Learning Multi-Class Dynamics,” Advances in Neural Information Processing Systems (NIPS '98), pages 389-395, 1998, proposes particle filters as an alternative to using linear models as the building blocks in a switching framework. The use of a nonparametric, particle-based model can be inefficient in domains where linear models are a powerful building block. Nonparametric methods are particularly expensive when applied to large state spaces, since they are exponential in the state space dimension.

In Brand, “Pattern discovery via entropy minimization”, Technical Report TR98-21, Mitsubishi Electric Research Lab, 1998, a Hidden Markov Model with an entropic prior is proposed for dynamics learning from sparse input data. The method is applied to the synthesis of facial animation. The dynamic models produced by this method are time invariant. Each state space neighborhood has a unique distribution over state transitions. In addition, the use of entropic priors results in fairly deterministic models learned from a moderate corpus of training data. In many applications time-invariant models are unlikely to succeed, since different state space trajectories can originate from the same starting point depending upon the class of motion being performed.

In Ghahramani et al., “Learning Nonlinear Stochastic Dynamics Using the Generalized EM Algorithm,” Advances in Neural Information Processing Systems (NIPS '99), pages 599-605, 1999, a Kalman smoother is used in conjunction with the generalized EM algorithm to learn a class of nonlinear dynamic models from input-output data. This approach also results in time-invariant models.

Another method addresses the problem of learning temporal models of motion in images. Unlike the more common state space models, this approach concentrates directly on the image space by representing any motion as a flow field in some particular flow field space. The basis of that space is learned from a corpus of examples. Hence, different bases capture distinct motion types. See Yacoob et al., “Learned Temporal Models of Image Motion,” Proceedings of Computer Vision and Pattern Recognition, pages 446-453, 1998.

One drawback of this method is that it only captures motion of a fairly fixed (and known) duration. For example, a prototypical walk of only one particular speed can be learned. Another disadvantage is that the models that result are highly viewpoint-specific, since they depend implicitly on the camera position. Furthermore, the approach is primarily suited for analysis rather than synthesis of motion sequences.

Technologies for analyzing the motion of the human figure play a key role in a broad range of applications, including computer graphics, user-interfaces, surveillance, and video editing. A motion of the figure can be represented as a trajectory in a state space which is defined by the kinematic degrees of freedom of the figure. Each point in state space represents a single configuration or pose of the figure. A motion such as a plié in ballet is described by a trajectory along which the joint angles of the legs and arms change continuously.

A key issue in human motion analysis and synthesis is modeling the dynamics of the figure. While the kinematics of the figure define the state space, the dynamics define which state trajectories are possible (or probable) in that state space. Prior methods for representing human dynamics have been based on analytic dynamic models. Analytic models are specified by a human designer. They are typically second order differential equations relating joint torque, mass, and acceleration.

The field of biomechanics is a source of complex and realistic analytic models of human dynamics. From the biomechanics point of view, the dynamics of the figure are the result of its mass distribution, joint torques produced by the motor control system, and reaction forces resulting from contact with the environment (e.g. the floor). Research efforts in biomechanics, rehabilitation, and sports medicine have resulted in complex, specialized models of human motion. For example, detailed walking models are described in Inman, Ralston and Todd, “Human Walking,” Williams and Wilkins, 1981.

The biomechanical approach has two drawbacks. First, the dynamics of the figure are quite complex, involving a large number of masses and applied torques, along with reaction forces which are difficult to measure. In principle, all of these factors must be modeled or estimated in order to produce physically-valid dynamics. Second, in some applications we may only be interested in a small set of motions, such as a vocabulary of gestures. In the biomechanical approach, it may be difficult to reduce the complexity of the model to exploit this restricted focus. Nonetheless, biomechanical models have been applied to human motion analysis.

A prior method for visual tracking uses a biomechanically-derived dynamic model of the upper body. See Wren and Pentland, “Dynamic models of human motion,” Proceeding of the Third Intermational Conference on Automatic Face and Gesture Recognition, pages 22-27, Nara, Japan, 1998. The unknown joint torques are estimated along with the state of the arms and head in an input estimation framework. A Hidden Markov Model is trained to represent plausible sequences of input torques. This prior art does not address the problem of modeling reaction forces between the figure and its environment. An example is the reaction force exerted by the floor on the soles of the feet during walking or running.

SUMMARY OF THE INVENTION

In view of the above discussion, therefore, there is a need for inference and learning methods for fully coupled SLDS models that can estimate a complete set of model parameters for a switching model given a training set of time-series data.

Described herein is a new class of approximate learning methods for switching linear dynamic (SLDS) models. These models consist of a set of linear dynamic system (LDS) models and a switching variable that indexes the active model. This new class has at least three advantages over dynamics learning methods known in the prior art:

New approximate inference techniques lead to tractable learning even when the set of LDS models is fully coupled.

The resulting models can represent time-varying dynamics, making them suitable for a wide range of applications.

All of the model parameters are learned from data, including the plant and noise parameters for the LDS models and Markov model parameters for the switching variable.

In addition, this method can be applied to the problem of learning dynamic models for human motion from data. It has at least three advantages over existing analytic dynamic models.

First, models can be constructed without a laborious manual process of specifying mass and force distributions. Moreover, it may be easier to tailor a model to a specific class of motion, as long as a sufficient number of samples are available.

Second, the same learning approach can be applied to a wide range of human motions from dancing to facial expressions.

Third, when training data is obtained from analysis of video measurements, the spatial and temporal resolution of the video camera determine the level of detail at which dynamical effects can be observed. Learning techniques can only model structure which is present in the training data. Thus, a learning approach is well-suited to building models at the correct level of resolution for video processing and synthesis.

A wide range of learning algorithms can be cast in the framework of Dynamic Bayesian Networks (DBNs). DBNs generalize two well-known signal modeling tools: Kalman filters for continuous state linear dynamic systems (LDS) and Hidden Markov Models (HMMs) for discrete state sequences. Kalman filters are described in Anderson et al., “Optimal Filtering,” Prentice-Hall, Inc., Englewood Cliffs, N.J., 1979. Hidden Markov Models are reviewed in Jelinek, “Statistical methods for speech recognition,” MIT Press, Cambridge, Mass., 1998.

Dynamic models learned from sequences of training data can be used to predict future values in a new sequence given the current values. They can be used to synthesize new data sequences that have the same characteristics as the training data. They can be used to classify sequences into different types, depending upon the conditions that produced the data.

We focus on a subclass of DBN models called Switching Linear Dynamics Systems. Intuitively, these models attempt to describe a complex nonlinear dynamic system with a succession of linear models that are indexed by a switching variable. The switching approach has an appealing simplicity and is naturally suited to the case where the dynamics are time-varying.

We present a method for approximate inference in fully coupled SLDS models. Exponentially hard exact inference is replaced with approximate inference of reduced complexity.

A first preferred embodiment uses Viterbi inference jointly in the switching and linear dynamic system states. A second preferred embodiment uses variational inference jointly in the switching and linear dynamic system states. A third preferred embodiment uses general pseudo Bayesian inference jointly in the switching and linear dynamic system states.

Parameters of a fully connected SLDS model are learned from data. Model parameters are estimated using a generalized expectation-maximization (EM) algorithm. Exact expectation/inference (E) step is replaced with one of the three approximate inference embodiments.

The learning method can be used to model the dynamics of human motion. The joint angles of the limbs and pose of the torso are represented as state variables in a switching linear dynamic model. The switching variable identifies a distinct motion regime within a particular type of human motion.

Motion regimes learned from figure motion data correspond to classes of human activity such as running, walking, etc. Inference produces a single sequence of switching modes which best describes a motion trajectory in the figure state space. This sequence segments the figure motion trajectory into motion regimes learned from data.

Accordingly, a method for determining, from a set of possible switching states and responsive to a sequence of measurements, a corresponding sequence of switching states for a system having a plurality of dynamic models, associates each model with a switching state such that a model is selected when its associated switching state is true. A state transition record is determined by determining and recording, for each possible switching state, an optimal prior switching state, based on the measurement sequence, where the optimal state optimizes a transition probability. For a final measurement, a most-probable final switching state is determined. Finally, the sequence of switching states is determined by backtracking, from the most-probable final switching state, through the state transition record.

Preferably, the switching states comprise a Markov chain.

In at least one embodiment, the transition probability is based on the likelihood of the measurement sequence, which may be determined with a Kalman filter, and the probability of transition between the switching states.

In at least one embodiment, the sequence is a time sequence. Each measurement corresponds to a distinct time.

In addition, at least one embodiment includes providing training data and learning parameters of the dynamic models in response to the determined sequence of switching states which result from the training data.

Future values in a new sequence may be predicted based on dynamic models learned from training sequences and the current values.

The sequence can comprise, for example, economic data, image or video data, audio data or spatial data. Audio data can include, for example, human speech, and more specifically, phonemes.

While the above embodiments are based on Viterbi techniques, other embodiments of the present invention are based on variational techniques. For example, a method for determining, from a set of possible switching states and responsive to a sequence of measurements, a corresponding sequence of switching states for a system having a plurality of dynamic models, includes defining a switching linear dynamic system (SLDS) having a plurality of dynamic models. Each dynamic model is associated with a switching state such that a dynamic model is selected when its associated switching state is true. The switching state at a particular instance is determined by a switching model, such as a hidden Markov model (HMM). The dynamic models are decoupled from the switching model, and parameters of the decoupled dynamic model are determined responsive to a switching state probability estimate. A state of a decoupled dynamic model corresponding to a measurement at the particular instance is estimated, responsive to one or more training sequences. Parameters of the decoupled switching model, which can include both input and output parameters, are then determined, responsive to the dynamic state estimate. A probability is estimated for each possible switching state of the decoupled switching model. The sequence of switching states is determined based on the estimated switching state probabilities.

Parameters of the dynamic models can be learned responsive to the determined sequence of switching states.

The most basic example of dynamics learning is system identification for a single linear model. This is a well-understood problem. See, for example, Ljung and Söderström, “Theory and Practice of Recursive Identification,” MIT Press, 1983. SLDS models and their equivalents have been studied in statistics, time-series modeling, and target tracking since the early 1970's. However, the complete learning framework we describe has never appeared in the literature.

Bar-Shalom and Li, “Estimation and tracking: principles, techniques, and software,” YBS, Storrs, Conn., 1998 and Kim, “Dynamic linear models with markov-switching,” Journal of Econometrics, 60:1-22, 1994, have developed a number of approximate pseudo-Bayesian inference techniques based on mixture component truncation or collapsing in SLDSs. They did not address the issue of learning system parameters. Shumway and Stoffer, “Dynamic linear models with switching,” Journal of the American Statistical Association, 86(415):763-769, September 1991, presented a systematic view of inference and learning in SLDS while assuming known prior switching state distributions at each time instance, Pr(s_(t))=π_(t)(i) and no temporal dependency between switching states. Krishnamurthy and Evans, “Finite-dimensional filters for passive tracking of markov jump linear systems,” Automatica, 34(6):765-770, 1998, imposed markov dynamics on the switching model. However, they assumed that noisy measurements of the switching states are available.

Ghahramani and Hinton, “Variational Learning for Switching State-Space Models,” submitted for publication in Neural Computation, 1998, to be published April, 2000, and incorporated herein by reference in its entirety, introduced a DBN-framework for learning and approximate inference in one class of SLDS models. Their underlying model differs from ours in assuming the presence of S independent, white noise-driven LDSs whose measurements are selected by the Markov switching process. Their assumption may lead to, among other things, measurements of processes that are, after some sufficient time, all in steady states and not changing. On the other hand, our model avoids this pitfall and may yield, if necessary, a dynamic set of measurements.

A switching model framework for particle filters is described in Isard and Blake, “A mixed-state CONDENSATION tracker with automatic model-switching,” Proceedings of International Conference on Computer Vision, pages 107-112, Bombay, India, 1998, and applied to dynamics learning in Blake, North and Isard, “Learning multi-class dynamics,” NIPS '98, 1998.

Manifold learning, as described by Bregler and Omohundro, “Nonlinear manifold learning for visual speech recognition,” Proceedings of International Conference on Computer Vision, pages 494-499, Cambridge, Mass., June 1995, is another approach to constraining the set of allowable trajectories within a high dimensional state space.

The learning framework of the present invention has at least three advantages over the alternative of manually deriving physically realistic dynamic models:

The first advantage is convenience. Models can be constructed without laborious manual process of specifying mass and force distributions. Moreover, it may be easier to tailor a model to a specific class of motion, as long as a sufficient number of samples are available.

A second advantage is resolution. The spatial and temporal resolution of a video source determine the level of detail at which dynamic effects can be observed. Since a learning technique can only model structure which is present in the visual signal, it is naturally tuned to building models at the correct level of resolution.

A third advantage is generality. The same learning approach can be applied to a wide range of human motions from dancing to facial expressions.

A primary advantage of our learning framework over previous learning techniques is the ability to learn time-varying models. Moreover, our framework is comprehensive in that all of the model parameters can be learned from data.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing and other objects, features and advantages of the invention will be apparent from the following more particular description of preferred embodiments of the invention, as illustrated in the accompanying drawings in which like reference characters refer to the same parts throughout the different views. The drawings are not necessarily to scale, emphasis instead being placed upon illustrating the principles of the invention.

FIG. 1 is a block diagram of a linear dynamical system, driven by white noise, whose state parameters are switched by a Markov chain model.

FIG. 2 is a dependency graph illustrating a fully coupled Bayesian network representation of the SLDS of FIG. 1.

FIG. 3 is a block diagram of an embodiment of the present invention based on approximate Viterbi inference.

FIGS. 4A and 4B comprise a flowchart illustrating the steps performed by the embodiment of FIG. 3.

FIG. 5 is a block diagram of an embodiment of the present invention based on approximate variational inference.

FIG. 6 is a dependency graph illustrating the decoupling of the hidden Markov model and SLDS in the embodiment of FIG. 5.

FIG. 7 is a flowchart illustrating the steps performed by the embodiment of FIG. 5.

FIGS. 8A and 8B comprise a flowchart illustrating the steps performed by a GPB2 embodiment of the present invention.

FIG. 9 comprises two graphs which illustrate learned segmentation of a “jog” motion sequence.

FIG. 10 is a block diagram illustrating classification of state space trajectories, as performed by the present invention.

FIG. 11 comprises several graphs which illustrate an example of segmentation.

FIG. 12 is a block diagram of a Kalman filter as employed by an embodiment of the present invention.

FIG. 13 is a diagram illustrating the operation of the embodiment of FIG. 12 for the specific case of figure tracking.

FIG. 14 is a diagram illustrating the mapping of templates.

FIG. 15 is a block diagram of an iterated extended Kalman filter (IEKF).

FIG. 16 is a block diagram of an embodiment of the present invention using an IEKF in which a subset of Viterbi predictions is selected and then updated.

FIG. 17 is a block diagram of an embodiment in which Viterbi predictions are first updated, after which a subset is selected.

FIG. 18 is a block diagram of an embodiment which combines SLDS prediction with sampling from a prior mixture density.

FIG. 19 is a block diagram of an embodiment in which Viterbi estimates are combined with updated samples drawn from a prior density.

FIG. 20 is a block diagram illustrating an embodiment of the present invention in which the framework for synthesis of state space trajectories in which an SLDS model is used as a generative model.

FIG. 21 is a dependency graph of an SLDS model, modified according to an embodiment of the present invention, with added continuous state constraints.

FIG. 22 is a dependency graph of an SLDS model, modified according to an embodiment of the present invention, with added switching state constraints.

FIG. 23 is a dependency graph of an SLDS model, modified according to an embodiment of the present invention, with both added continuous and switching state constraints.

FIG. 24 is a block diagram of the framework for a synthesis embodiment of the present invention using constraints and utilizing optimal control.

FIG. 25 is an illustration of a stick figure motion sequence as synthesized by an embodiment of the present invention.

DETAILED DESCRIPTION OF THE INVENTION SWITCHING LINEAR DYNAMIC SYSTEM MODEL

FIG. 1 is a block diagram of a complex physical linear dynamic system (LDS) 10, driven by white noise v_(k), also called “plant noise.” The LDS state parameters evolve in time according to some known model, such as a Markov chain (MC) model 12.

The system can be described using the following set of state-space equations:

x _(t+1) =A(s _(t+1))x _(t) +v _(t+1)(s _(t+1)),

y _(t) =Cx _(t) +w _(t), and

x ₀ =v ₀(s ₀)  (Eq. 1)

for the physical system 10, and

Pr(s _(t+1) |s _(t))=s′ _(t+1) Πs _(t), and

Pr(s ₀)=π₀

for the switching model 12.

Here, x_(t) ε denotes the hidden state of the LDS 10 at time t, and v_(t) is the state noise process. Similarly, y_(t) ε is the observed measurement at time t, and w_(t) is the measurement noise. Parameters A and C are the typical LDS parameters: the state transition matrix 14 and the observation matrix 16, respectively. Assuming the LDS models a Gauss-Markov process, the noise processes are independently distributed Gaussian:

v _(t))(s _(t))˜N(0,Q(s _(t))),t>0

v ₀)(s ₀)˜N(x ₀(s ₀),Q ₀(s ₀))

w _(t) ˜N(0,R).

where Q is the state noise variance and R is the measurement noise variance. The notation s′ is used to indicate the transpose of vector s.

The switching model 12 is assumed to be a discrete first-order Markov process. State variables of this model are written as s_(t). They belong to the set of S discrete symbols {e₀, . . . , e_(S−1)}, where e_(i) is, for example, the unit vector of dimension S with a non-zero element in the i-th position. The switching model 12 is a first-order discrete Markov model defined with the state transition matrix Π whose elements are

Π(i,j)=Pr(s _(t+1) =e _(i) |s _(t) =e _(j)),  (Eq. 2)

and which is given an initial state distribution Π₀.

Coupling between the LDS and the switching process is full and stems from the dependency of the LDS parameters A and Q on the switching process state s_(t). Namely,

A(s _(t) =e _(i))=A _(i)

Q(s _(t) =e _(i))=Q _(i)

In other words, switching state s_(t) determines which of S possible models {(A₀,Q₀), . . . , (A_(S−1),Q_(S−1))} is used at time t.

FIG. 2 is a dependency graph 20 equivalently illustrating a rather simple but fully coupled Bayesian network representation of the SLDS, where each s_(t) denotes an instance of one of the discrete valued action states which switch the physical system models having continuous valued states x and producing observations y. The full model can be written as the “joint distribution” P: ${P\left( {Y_{T},X_{T},S_{T}} \right)} = {{\Pr \left( s_{0} \right)}{\prod\limits_{t = 1}^{T - 1}\quad {\Pr \left( s_{t} \middle| s_{t - 1} \right)}}}$ ${\Pr \left( x_{0} \middle| s_{0} \right)}{\prod\limits_{t = 1}^{T - 1}\quad {\Pr \left( {\left. x_{t} \middle| x_{t - 1} \right.,s_{t}} \right){\prod\limits_{t = 0}^{T - 1}\quad {{\Pr \left( y_{t} \middle| x_{t} \right)}.}}}}$

where Y_(T), X_(T), and S_(T) denote the sequences, of length T, of observation and hidden state variables, and switching variables, respectively. For example, Y_(T)={y₀, . . . , Y_(T−1)}. In this dependency graph 20, the coupling between the switching states and the LDS states is full, i.e., switching states are time-dependent, LDS states are time-dependent, and switching and LDS states are intradependent.

We can now write an equivalent representation of the fully coupled SLDS as the above DBN, assuming that the necessary conditional probability distribution functions (pdfs) are appropriately defined. Assuming a Gauss-Markov process on the LDS, it follows:

x _(t+1) |x _(t) ,s _(t+1) =e _(i) ˜N(A _(i) x _(t) ,Q _(i)),

y _(t) |x _(t) ˜N(Cx _(t) ,R),

x ₀ |s ₀ =e _(i) ˜N(x _(0,i) ,Q ₀ ,i)

Recalling the Markov switching model assumption, the joint pdf of the complex DBN of duration T, or, equivalently, its Hamiltonian, where the Hamiltonian H(x) of a distribution P(x) is defined as any positive function such that P(x)=[(exp(−H(x)))/(Σ_(ψ)exp(−H(ψ)))], can be written as: $\begin{matrix} \begin{matrix} {{H\left( {X_{T},S_{T},Y_{T}} \right)} = \quad {\frac{1}{2}{\sum\limits_{t = 1}^{T - 1}\quad {\sum\limits_{i = 0}^{N - 1}\quad \left\lbrack {{\left( {x_{t} - {A_{i}x_{t - 1}}} \right)^{\prime}{Q_{i}^{- 1}\left( {x_{t} - {A_{i}x_{t - 1}}} \right)}} +} \right.}}}} \\ {{{\left. {{\quad \log \quad }Q_{i}} \right\rbrack }{s_{t}(i)}} + {\frac{1}{2}{\sum\limits_{i = 0}^{N - 1}\left\lbrack {{\left( x_{0,i} \right)^{\prime}{Q_{0,i}^{- 1}\left( x_{0,i} \right)}} +} \right.}}} \\ {{\left. \quad {\log \quad {Q_{0,i}}} \right\rbrack {s_{0}(i)}} + {\frac{NT}{2}\log \quad 2\quad \pi} + {\frac{1}{2}{\sum\limits_{t = 0}^{T - 1}\left( {y_{t} - {Cx}_{t}} \right)^{\prime}}}} \\ {\quad {{R^{- 1}\left( {y_{t} - {Cx}_{t}} \right)} + {\frac{T}{2}\log \quad {R}} + {\frac{MT}{2}\log \quad 2\quad \pi} +}} \\ {\quad {{\sum\limits_{t - 1}^{T - 1}\quad {{s_{t}^{\prime}\left( {{- \log}\quad \Pi} \right)}s_{t - 1}}} + {{s_{0}^{\prime}\left( {{- \log}\quad \pi_{0}} \right)}.}}} \end{matrix} & \left( {{Eq}.\quad 3} \right) \end{matrix}$

INFERENCE

The goal of inference in SLDSs is to estimate the posterior probability of the hidden states of the system (s_(t) and x_(t)) given some known sequence of observations Y_(T) and the known model parameters, i.e., the likelihood of a sequence of models as well as the estimates of states. Namely, we need to find the posterior

P(X _(T) ,S _(T) |Y _(T))=Pr(X _(T) ,S _(T) |Y _(T)),

or, equivalently, its “sufficient statistics”. Given the form of P it is easy to show that these are the first and second-order statistics: mean and covariance among hidden states x_(t),x_(t−1),s_(t),s_(t−1).

If there were no switching dynamics, the inference would be straightforward—we could infer X_(T) from Y_(T) using an LDS inference approach such as smoothing, as described by Rauch, “Solutions to the linear smoothing problem,” IEEE Trans. Automatic Control, AC-8(4):371-372, October 1963. However, the presence of switching dynamics embedded in matrix P makes exact inference more complicated. To see that, assume that the initial distribution of x₀ at t=0 is Gaussian. At t=1, the pdf of the physical system state x₁ becomes a mixture of S Gaussian pdfs since we need to marginalize over S possible but unknown models. At time t, we will have a mixture of S^(t) Gaussians, which is clearly intractable for even moderate sequence lengths. It is therefore necessary to explore approximate inference techniques that will result in a tractable learning method. What follows are three preferred embodiments of the inference step.

APPROXIMATE VITERBI INFERENCE EMBODIMENT

FIG. 3 is a block diagram of an embodiment of a dynamics learning method based on approximate Viterbi inference. At step 32, switching dynamics of a number of SLDS motion models are learned from a corpus of state space motion examples 30. Parameters 36 of each SLDS are re-estimated, at step 34, iteratively so as to minimize the modeling cost of current state sequence estimates, obtained using the approximate Viterbi inference procedure. Approximate Viterbi inference is developed as an alternative to computationally expensive exact state sequence estimation.

The task of the Viterbi approximation approach of the present invention is to find the most likely sequence of switching states s_(t) for a given observation sequence Y_(T). If the best sequence of switching states is denoted S_(T)*, then the desired posterior

P(X _(T) ,S _(T) |Y _(T)) can be approximated as

P(X _(T) ,S _(T) |Y _(T))=P(X _(T) |S _(T) ,Y _(T))P(S _(T) |Y _(T))≈P(X _(T) |S _(T) ,Y _(T))δ(S _(T) −S _(T)*),  (Eq. 4)

where δ(x)=1 if x=0 and δ(x)=0 if x≠0. In other words, the switching sequence posterior P(S_(T)|Y_(T)) is approximated by its mode. Applying Viterbi inference to two simpler classes of models, discrete state hidden Markov models and continuous state Gauss-Markov models is well-known. An embodiment of the present invention utilizes an algorithm for approximate Viterbi inference that generalizes the two approaches.

We would like to determine the switching sequence S_(T)* such that S_(T)*=arg max_(ST)P(S_(T)|Y_(T)). First, define the following probability J_(t,i) up to time t of the switching state sequence being in state i at time t given the measurement sequence Y_(t): $\begin{matrix} {J_{t,i} = {\max\limits_{S_{t - 1}}\quad {P\left( {S_{t - 1},{s_{t} = e_{i}},Y_{t}} \right)}}} & \left( {{Eq}.\quad 5} \right) \end{matrix}$

If this quantity is known at time T, the probability of the most likely switching sequence S_(T)* is simply P(S_(T)*|Y_(T))∝max_(i)J_(T−1,i). In fact, a recursive procedure can be used to obtain the desired quantity. To see that, express J_(t,i) in terms of J_(s) at t−1. It follows that $\begin{matrix} \begin{matrix} {J_{t,i} = \quad {\max\limits_{S_{t - 1}}{P\left( {S_{t - 1},{s_{t} = e_{i}},Y_{t}} \right)}}} \\ {= \quad {\underset{S_{t - 1}}{\max \quad}{P\left( {S_{t - 1},{s_{t} = e_{i}},Y_{t - 1},y_{t}} \right)}}} \\ {= \quad {\max\limits_{S_{t - 1}}{{P\left( {\left. y_{t} \middle| S_{t - 1} \right.,{s_{t} = e_{i}},Y_{t - 1}} \right)}{P\left( {s_{t} = \left. e_{i} \middle| {S_{{t - 1},}Y_{t - 1}} \right.} \right)}}}} \\ {\quad {P\left( {S_{{t - 1},}Y_{t - 1}} \right)}} \\ {\approx \quad {\max\limits_{j}\left\{ {P\left( {{\left. y_{t} \middle| s_{t} \right. = e_{i}},{s_{t - 1} = e_{j}},{S_{t - 2}^{*}(j)},Y_{t - 1}} \right)} \right.}} \\ {\quad {{P\left( {s_{t} = {\left. e_{i} \middle| s_{t - 1} \right. = e_{j}}} \right)}{\max\limits_{S_{t - 2}}\quad \left. {P\left( {S_{t - 2},{s_{t - 1} = e_{j}},Y_{t - 1}} \right)} \right\}}}} \\ {= \quad {\max\limits_{j}\left\{ {J_{{t|{t - 1}},i,j}J_{{t - 1},j}} \right\}}} \end{matrix} & \left( {{Eq}.\quad 6} \right) \end{matrix}$

where we denote

J _(t|t−1,i,j) =P(y _(t) |s _(t) =e _(i) ,s _(t−1) =e _(j) ,S _(t−2)*(j),Y _(t−1))P(s _(t) =e _(i) |s _(t−1) =e _(j))  (7)

as the “transition probability” from state j at time t−1 to state i at time t. Since this analysis takes place relative to time t, we refer to s_(t) as the “switching state,” and to s_(t−1) as the “previous switching state.” S_(t−2)*(i) is the “best” switching sequence up to time t−1 when SLDS is in state i at time t−1: $\begin{matrix} {{S_{t - 2}^{*}(i)} = {\arg \quad {\max\limits_{S_{t - 2}}{J_{{t - 1},i}.}}}} & \left( {{Eq}.\quad 8} \right) \end{matrix}$

Hence, the switching sequence posterior at time t can be recursively computed from the same at time t−1. The two scaling components in J_(t|t−1,i,j) are the likelihood associated with the transition j→i from t−1 to t, and the probability of discrete SLDS switching from j to i.

To find the likelihood term, note that concurrently with the recursion of Equation 6, for each pair of consecutive switching states j,i at times t−1,t, one can obtain the following statistics using the Kalman filter: ${\hat{X}}_{{t|t},i}\overset{\Delta}{=}{\langle{\left. x_{t} \middle| Y_{t} \right.,{s_{t} = e_{i}}}\rangle}$ $\Sigma_{{t|t},i}\overset{\Delta}{=}{\langle{\left. {\left( {x_{t} - {\hat{x}}_{{t|t},i}} \right)\left( {x_{t} - {\hat{x}}_{{t|t},i}} \right)^{\prime}} \middle| Y_{t} \right.,{s_{t} = e_{i}}}\rangle}$ ${\hat{x}}_{{t|{t - 1}},i,j}\overset{\Delta}{=}{\langle{\left. x_{t} \middle| Y_{t - 1} \right.,{s_{t} = e_{i}},{s_{t - 1} = e_{j}}}\rangle}$ $\Sigma_{{t|{t - 1}},i,j}\overset{\Delta}{=}{\langle{\left. {\left( {x_{t} - {\hat{x}}_{{t|t},i,j}} \right)\left( {x_{t - 1} - {\hat{x}}_{{t|{t - 1}},i,j}} \right)^{\prime}} \middle| Y_{t} \right.,{s_{t} = e_{i}},{s_{t - 1} = e_{j}}}\rangle}$ ${\hat{x}}_{{t|t},i,j}\overset{\Delta}{=}{\langle{\left. x_{t} \middle| Y_{t} \right.,{s_{t} = e_{i}},{s_{t - 1} = e_{j}}}\rangle}$ $\Sigma_{{t|t},i,j}\overset{\Delta}{=}{\langle{\left. {\left( {x_{t} - {\hat{x}}_{{t|t},i}} \right)\left( {x_{t} - {\hat{x}}_{{t|t},j}} \right)^{\prime}} \middle| Y_{t} \right.,{s_{t} = e_{i}},{s_{t - 1} = e_{j}}}\rangle}$

where {circumflex over (x)}_(t|t,i) is the “best” filtered LDS state estimate at t when the switch is in state i at time t and a sequence of t measurements, Y_(t), has been processed; {circumflex over (x)}_(t|t−1,i,j) and {circumflex over (x)}_(t|t,i,j) are the one-step predicted LDS state and the “best” filtered state estimates at time t, respectively, given that the switch is in state i at time t and in state j at time t−1 and only t−1 measurements are known. The two sets {{circumflex over (x)}_(t|t−1,i,j)} and {{circumflex over (x)}_(t|t,i,j)}, where i and j take on all possible values, are examples of “sets of continuous state estimates.” The set {{circumflex over (x)}_(t|t−1,i,j)} is obtained through “Viterbi prediction.” Similar definitions are easily obtained for filtered and predicted state variance estimates, Σ_(t|t,i) and Σ_(t|t−1,i,j) respectively. For a given switch state transition j→i it is now easy to establish relationship between the filtered and the predicted estimates. From the theory of Kalman estimation, it follows that for transition j→i the following time updates hold:

 {circumflex over (x)} _(t|t−1,i,j) =A _(i) {circumflex over (x)} _(t−1|t−1,j)  (Eq. 9)

Σ_(t|t−1,i,j) =A _(i)Σ_(t−1|t−1,j) A′ _(i) +Q _(i)  (Eq. 10)

Given a new observation y_(t) at time t, each of these predicted estimates can now be filtered using a Kalman “measurement update” framework. For instance, the state estimate measurement update equation yields

{circumflex over (x)} _(t|t,i,j) ={circumflex over (x)} _(t|t−1,i,j) +K _(i,j)(y _(t) −C{circumflex over (x)} _(t|t−1,i,j)).  (Eq. 11).

where K_(i,j) is the Kalman gain matrix associated with the transitions j→i.

Appropriate equations can be obtained that link Σ_(t|t−1,i,j) and Σ_(t|t,i,j). The likelihood term can then be easily computed as the probability of innovation y_(t)−C{circumflex over (x)}_(t|t−1,i,j) of j→i transition,

y _(t) |s _(t) =e _(i) ,s _(t−1) =e _(j) ,S _(t−2)*(j)˜N(y _(t) ;C{circumflex over (x)} _(t|t−1,i,j) ,CΣ _(t|t−1,i,j) C′+R)  (Eq. 12)

Obviously, for every current switching state i there are S possible previous switching states. To maximize the overall probability at every time step and for every switching state, one “best” previous state j is selected:

ψ_(t−1,i)=arg max_(j) {J _(t|t−1,i,j) J _(t−1,j)}  (Eq. 13)

Since the best state is selected based on the continuous state predictions from the previous time step, ψ_(t−1,i) is referred to as the “optimal prior switching state.” The index of this state is kept in the state transition record entry ψ_(t−1,i). Consequently, we now obtain a set of S best filtered LDS states and variances at time t:

{circumflex over (x)} _(t|t,i) ={circumflex over (x)} _(t|t,i,ψ) _(t−1,i) and Σ_(t|t,i)=Σ_(t|t,i,ψ) _(t−1,i) .

Once all T observations Y_(T−1) have been fused to decode the “best” switching state sequence, one uses the index of the best final state, i_(T−1)*=arg max_(j)J_(T−1,i), and then traces back through the state transition record ψ_(t−1,i), setting i_(t)*=ψ_(t,i) _(t+1) _(*). The switching model's sufficient statistics are now simply <s_(t)>=e_(i) _(t) _(*) and <s_(t)s′_(t−1)>=e_(i) _(t) _(*)e′_(i) _(t−1) _(*).

Given the “best” switching state sequence, the sufficient LDS statistics can be easily obtained using Rauch-{dot over (T)}ung-Streiber (RTS) smoothing. Smoothing is described in Anderson et al, “Optimal Filtering,” Prentice-Hall, Inc., Englewood Cliffs, N.J., 1979. For example, ${\langle{x_{t},{s_{t}(i)}}\rangle} = \left\{ \quad \begin{matrix} {\hat{x}}_{{t|{T - 1}},i_{j}^{*}} & {i = i_{t}^{*}} \\ 0 & {otherwise} \end{matrix}\quad \right.$

for i=0, . . . , S−1.

FIGS. 4A and 4B comprise a flowchart that summarizes an embodiment of the present invention employing the Viterbi inference algorithm for SLDSs, as described above. The steps are as follows:

Initialize LDS state estimates {circumflex over (χ)}_(0|−1,i) and Σ_(0|−1,i); (Step 102) Initialize J_(0,i). (Step 102) for i = 1:T−1 (Steps 104, 122) for j = 1:S (Steps 106, 120) for j = 1:S (Steps 108, 114) Predict and filter LDS state estimates (Step 110) {circumflex over (χ)}_(t|t,i,j) and Σ_(t|t,i,j) Find j → i “transition probability” J_(t|t−1,i,j) (Step 112) end Find best transition J_(t,i), into state i; (Step 116) Update sequence probabilities J_(t,i) and LDS state (Step 118) estimates {circumflex over (χ)}_(t|t,i) and Σ_(t|t,i) end Find “best” final switching state i_(T−1) ^(*) (Step 124) Backtrack to find “best” switching state sequence i_(t) ^(*) (Step 126) Find DBN's sufficient statistics. (Step 128)

APPROXIMATE VARIATIONAL INFERENCE EMBODIMENT

FIG. 5 is a block diagram for an embodiment of the dynamics learning method based on approximate variational inference. Generally, reference numbers 40-46 correspond to reference numbers 30-36 of FIG. 3, the approximate Viterbi inference block 34 of FIG. 3 being replaced by the approximate variational inference block 44.

At step 42, the switching dynamics of one or more SLDS motion models are learned from a corpus of state space motion examples 40. Parameters 46 of each SLDS are re-estimated, at step 44, iteratively so as to minimize the modeling cost of current state sequence estimates that are obtained using the approximate variational inference procedure. Approximate variational inference is developed as an alternative to computationally expensive exact state sequence estimation.

A general structured variational inference technique for Bayesian networks is described in Jordan at al., “An Introduction to Variational Methods For Graphical Models,” Learning In Graphical Models, Kluwer Academic Publishers, 1998. They consider a parameterized distribution Q which is in some sense close to the desired conditional distribution P, but which is easier to compute. Q can then be employed as an approximation of P,

P(X _(T) ,S _(T) |Y _(T))≈Q(X _(T) ,S _(T) |Y _(T))

Namely, for a given set of observations Y_(T), a distribution Q(X_(T),S_(T)|η,Y_(T) with an additional set of variational parameters h is defined such that Kullback-Leibler divergence between Q(X_(T),S_(T)|η,Y_(T) and P(X_(T),S_(T)|,Y_(T)) is minimized with respect to h: $\eta^{*} = {\arg \quad {\min\limits_{\eta}\quad {\sum\limits_{S_{t}}\quad {\int_{X_{T}}{Q\left( {X_{T},\left. S_{T} \middle| \eta \right.,{Y_{T}\log \quad {\frac{P\left( {X_{T},\left. S_{T} \middle| Y_{T} \right.} \right)}{Q\left( {X_{T},\left. S_{T} \middle| \eta \right.,Y_{T}} \right.}.}}} \right.}}}}}$

The dependency structure of Q is chosen such that it closely resembles the dependency structure of the original distribution P. However, unlike P, the dependency structure of Q must allow a computationally efficient inference. In our case, we define Q by decoupling the switching and LDS portions of SLDS as shown in FIG. 6.

FIG. 6 illustrates the factorization of the original SLDS. The two subgraphs of the original network are a Hidden Markov Model (HMM) Q_(S) 50 with variational parameters {q₀, . . . , q_(T−1),} and a time-varying LDS. Q_(X) 52 with variational parameters {{circumflex over (x)}₀, Â₀, . . . , Â_(T−1), {circumflex over (Q)}₀, . . . , {circumflex over (Q)}_(T−1)}. More precisely, the Hamiltonian of the approximating dependency graph is defined as: $\begin{matrix} \begin{matrix} {{H_{Q}\left( {X_{T},S_{T},Y_{T}} \right)} = \quad {{\frac{1}{2}{\sum\limits_{t = 1}^{T - 1}{\left( {x_{t} - {{\hat{A}}_{t}x_{t - 1}}} \right)^{\prime}{{\hat{Q}}_{t}^{- 1}\left( {x_{t} - {{\hat{A}}_{t}x_{t - 1}}} \right)}}}} +}} \\ {\quad {{\frac{1}{2}\log \quad {{\hat{Q}}_{t}}} + {\frac{1}{2}\left( {x_{0} - {\hat{x}}_{0}} \right)^{\prime}{{\hat{Q}}_{0}^{- 1}\left( {x_{0} - {\hat{x}}_{0}} \right)}} +}} \\ {\quad {{\frac{1}{2}\log \quad {{\hat{Q}}_{0}}} + {\frac{NT}{2}\log \quad 2\quad \pi} + {\frac{1}{2}{\sum\limits_{t = 0}^{T - 1}\left( {y_{t} - {Cx}_{t}} \right)^{\prime}}}}} \\ {\quad {{R^{- 1}\left( {y_{t} - {Cx}_{t}} \right)} + {\frac{T}{2}\log \quad {R}} + {\frac{MT}{2}\log \quad 2\quad \pi} +}} \\ {\quad {{\sum\limits_{t = 1}^{T - 1}{{s_{t}^{\prime}\left( {{- \log}\quad \Pi} \right)}s_{t - 1}}} + {s_{0}^{\prime}\left( {{- \log}\quad \pi_{0}} \right)} +}} \\ {\quad {\sum\limits_{t = 1}^{T - 1}{{s_{t}^{\prime}\left( {{- \log}\quad q_{t}} \right)}.}}} \end{matrix} & \left( {{Eq}.\quad 14} \right) \end{matrix}$

The two subgraphs 50, 52 are “decoupled,” thus allowing for independent inference, Q(X_(T),S_(T)|η,Y_(T))=Q(X_(T)|η,Y_(T))Q_(S)(S_(T)|η). This is also reflected in the sufficient statistics of the posterior defined by the approximating network, e.g., <x_(t)x′_(t)s_(t)>=<x_(t)x′_(t)><s_(t)>.

The optimal values of the variational parameters h are obtained by setting the derivative of the KL-divergence with respect to h to zero. We can then arrive at the following optimal variational parameters: $\begin{matrix} {{{\hat{Q}}_{T - 1}^{- 1} = {\sum\limits_{i = 0}^{S - 1}\quad {Q_{i}^{- 1}{\langle{s_{t}(i)}\rangle}}}}\begin{matrix} {{\hat{Q}}_{t}^{- 1} = \quad {{\sum\limits_{i = 0}^{S - 1}\quad {Q_{i}^{- 1}{\langle{s_{t}(i)}\rangle}}} + {\sum\limits_{i = 0}^{S - 1}{A_{i}^{\prime}Q_{i}^{- 1}{\langle{s_{t + 1}(i)}\rangle}}} -}} \\ {\quad {{{\hat{A}}_{t + 1}^{\prime}{\hat{Q}}_{t + 1}^{- 1}A_{t + 1}},{0 < t < {T - 1}}}} \end{matrix}{{\hat{Q}}_{0}^{- 1} = {{\sum\limits_{i = 0}^{S - 1}\quad {Q_{0,i}^{- 1}{\langle{s_{0}(i)}\rangle}}} + {\sum\limits_{i = 0}^{S - 1}{A_{i}^{\prime}Q_{i}^{- 1}A_{i}{\langle{s_{1}(i)}\rangle}}} - {{\hat{A}}_{1}^{\prime}{\hat{Q}}_{1}^{- 1}{\hat{A}}_{1}}}}{{\hat{A}}_{t} = {{\hat{Q}}_{t}{\sum\limits_{i = 0}^{S - 1}\quad {Q_{i}^{- 1}A_{i}{\langle{s_{t}(i)}\rangle}}}}}{{\hat{x}}_{0} = {{\hat{Q}}_{0}{\sum\limits_{i = 0}^{S - 1}\quad {Q_{0,i}^{- 1}x_{0,i}{\langle{s_{0}(i)}\rangle}}}}}} & \left( {{Eq}.\quad 15} \right) \\ {{{\log \quad {q_{0}(i)}} = {{{- \frac{1}{2}}{\langle{\left( {x_{0} - x_{0,i}} \right)^{\prime}{{\hat{Q}}_{0,i}^{- 1}\left( {x_{0} - x_{0,i}} \right)}}\rangle}} - {\frac{1}{2}\log \quad {{\hat{Q}}_{0,i}}}}}\begin{matrix} {{\log \quad {q_{t}(i)}} = \quad {{{- \frac{1}{2}}{\langle{\left( {x_{t} - {A_{i}x_{t - 1}}} \right)^{\prime}{{\hat{Q}}_{i}^{- 1}\left( {x_{t} - {A_{i}x_{t - 1}}} \right)}}\rangle}} -}} \\ {\quad {{\frac{1}{2}\log {{\hat{Q}}_{t,i}}},{t > 0}}} \end{matrix}} & \left( {{Eq}.\quad 16} \right) \end{matrix}$

To obtain the expectation terms <s_(t)>=Pr(s_(t)|q₀, . . . ,q_(T−1)), we use the inference in the HMM with output “probabilities” q_(t), as described in Rabiner and Juang, “Fundamentals of Speech Recognition,” Prentice Hall, 1993. Similarly, to obtain <x_(t)>=E[x_(t)|Y_(T)], LDS inference is performed in the decoupled time-varying LDS via RTS smoothing. Since Â_(t),{circumflex over (Q)}_(t) in the decoupled LDS Q_(X) depend on <s_(t)> from the decoupled HMM Q_(S), and q_(t) depends on <x_(t)>,<x_(t)x′_(t)>,<x_(t)x′_(t−1)> from the decoupled LDS, Equations 15 and 16 together with the inference solutions in the decoupled models form a set of fixed-point equations. Solution of this fixed-point set yields a tractable approximation to the intractable inference of the original fully coupled SLDS.

FIG. 7 is a flowchart summarizing the variational inference algorithm for fully coupled SLDSs, corresponding to the steps below:

error = ∞; (Step 152) Initialize <s_(t)>; (Step 152) while (error > maxError) { (Step 154) Find {circumflex over (Q)}₁, Â_(t), {circumflex over (x)}₀ from <s_(t)> using Equations 11; (Step 156) Estimate <x_(t)>,<x_(t)x′_(t)> and <x_(t)x′_(t-1)> from Y_(T) using time-varying LDS inference; (Step 158) Find q_(t) from <x_(t)>,<x_(t)x′_(t)> and <x_(t)x′_(t-1)> using Equations 12; (Step 160) Estimate <s_(t)> from q₁ using HMM inference. (Step 162) Update approximation error (KL divergence); (Step 164) }

Variational parameters in Equations 15 and 16 have an intuitive interpretation. Variational parameters of the decoupled LDS {circumflex over (Q)}_(t) and Â_(t) in Equation 15 define a best unimodal (non-switching) representation of the corresponding switching system and are, approximately, averages of the original parameters weighted by a best estimates of the switching states P(s). Variational parameters of the decoupled HMM log q_(t)(i) in Equation 16 measure the agreement of each individual LDS with the data.

GENERAL PSEUDO BAYESIAN INFERENCE EMBODIMENT

The Generalized Pseudo Bayesian (GPB) approximation scheme first introduced in Bar-Shalom et al., “Estimation and Tracking: Principles, Techniques, and Software,” Artech House, Inc. 1993 and in Kim, “Dynamic Linear Models With Markov-Switching,” Journal of Econometrics, volume 60, pages 1-22, 1994, is based on the idea of “collapsing”, i.e., representing a mixture of M^(t) Gaussians with a mixture of M^(T) Gaussians, where r<t. A detailed review of a family of similar schemes appears in Kevin P. Murphy, “Switching Kalman Filters,” Technical Report 98-10, Compaq Cambridge Research Lab, 1998. We develop a SLDS inference scheme jointly in the switching and linear dynamic system states, derived from filtering GPB2 approach of Bar-Shalom et al. The algorithm maintains a mixture of M² Gaussians over all times.

GPB2 is closely related to the Viterbi approximation described previously. It differs in that instead of picking the most likely previous switching state j at every time step t and switching state i, we collapse the M Gaussians (one for each possible value of j) into a single Gaussian.

Consider the filtered and predicted means {circumflex over (x)}_(t|t,i,j) and {circumflex over (x)}_(t|t−1,i,j), and their associated covariances, which were defined previously with respect to the approximate Viterbi embodiment. Assume that for each switching state i and pairs of states (i,j) the following distributions are defined at each time step:

Pr(s _(t) =i|Y _(t))

Pr(s _(t) =i,s _(t−1) =j|Y _(t))

Regular Kalman filtering update can be used to fuse the new measurement y_(t) and obtain S² new SLDS states at t for each S states at time t−1, in a similar fashion to the Viterbi approximation embodiment discussed previously.

Unlike Viterbi approximation which picks one best switching transition for each state i at time t, GPB2 “averages” over S possible transitions from t−1. Namely, it is easy to see that

Pr(s _(t) =i,s _(t−1) =j|Y _(t))≈Pr(y _(t) |{circumflex over (x)} _(t,i,j))Π(i,j)Pr(s _(t−1) =j|Y _(t−1)).

From there it follows immediately that the current distribution over the switching states is ${\Pr \left( {s_{t} = \left. i \middle| Y_{t} \right.} \right)} = {\sum\limits_{j}{\Pr \left( {{s_{t} = i},{s_{t - 1} = \left. j \middle| Y_{t} \right.}} \right)}}$

and that each previous state j now has the following posterior ${\Pr \left( {{s_{t - 1} = {\left. j \middle| s_{t} \right. = i}},Y_{t}} \right)} = {\frac{\Pr \left( {{s_{t} = i},{s_{t - 1} = \left. j \middle| Y_{t} \right.}} \right)}{\Pr \left( {s_{t} = \left. i \middle| Y_{t} \right.} \right)}.}$

This posterior is important because it allows one to “collapse” or “average” the S transitions into each state i into one average state, e.g. ${\hat{x}}_{{t|t},i} = {\sum\limits_{j}{{\hat{x}}_{{t|t},i,j}{{\Pr \left( {{s_{t - 1} = {\left. j \middle| s_{t} \right. = i}},Y_{t}} \right)}.}}}$

Analogous expressions can be obtained for the variances S_(t|t,i) and S_(t,t−1|t,i).

Smoothing in GPB2 is a more involved process that includes several additional approximations. Details can be found in Kevin P. Murphy, “Switching Kalman Filters,” Technical Report 98-10, Compaq Cambridge Research Lab, 1998. First an assumption is made that decouples the switching model from the LDS when smoothing the switching states. Smoothed switching states are obtained directly from Pr(s_(t)|Y_(t)) estimates, namely Pr(s_(t)=i|s_(t+1)=k,Y_(T))≈Pr(s_(t)=i|s_(t+1)=k,Y_(t)). Additionally, it is assumed that {circumflex over (x)}_(t+1|T,i,k)≈{circumflex over (x)}_(t+1|T,k). These two assumptions lead to a set of smoothing equations for each transition (i,k) from t to t+1 that obey an RTS smoother, followed by collapsing similar to the filtering step.

FIGS. 8A and 8B comprise a flowchart illustrating the steps employed by a GPB2 embodiment, as summarized by the following steps:

Initialize LDS state estimates {circumflex over (χ)}_(0|−1,i) and Σ_(0|−1,i); (Step 202) Initialize Pr(s₀ = i) = p(i), for i=0, . . . , S−1; (Step 202) for t = 1:T−1 (Steps 204, 222) for i = 1:S (Steps 206, 220) for j = 1:S (Steps 208, 216) Predict and filter LDS state estimates {circumflex over (χ)}_(t|t,i,j), Σ_(t|t,i,j); (Step 210) Finding switching state distributions (Step 212) Pr(s_(t) = i|Y_(t)), Pr(s_(t−1) = j|s_(t) = i,Y_(t)); Collapse {circumflex over (χ)}_(t|t,i,j), Σ_(t|t,i,j) to {circumflex over (χ)}_(t|t,i), Σ_(t|t,i); (Step 214) end Collapse {circumflex over (χ)}_(t|t,i) and Σ_(t|t,i) to {circumflex over (χ)}_(t|t) and Σ_(t|t,i). (Step 218) end end Do GPB2 smoothing; (Step 224) Find sufficient statistics. (Step 226)

The inference process of the GPB2 embodiment is clearly more involved than those of the Viterbi or the variational approximation embodiments. Unlike Viterbi, GPB2 provides soft estimates of switching states at each time t. Like Viterbi, GPB2 is a local approximation scheme and as such does not guarantee global optimality inherent in the variational approximation. However, Xavier Boyen and Daphne Koller, “Tractable inference for complex stochastic processes,” Uncertainty in Artificial Intelligence, pages 33-42, Madison, Wis., 1998, provides complex conditions for a similar type of approximation in general DBNs that lead to globally optimal solutions.

LEARNING OF SLDS PARAMETERS

Learning in SLDSs can be formulated as the problem of maximum likelihood learning in general Bayesian networks. Hence, a generalized Expectation-Maximization (EM) algorithm can be used to find optimal values of SLDS parameters {A₀, . . . , A_(s−1),C,Q₀, . . . , Q_(s−1),R,Π,π₀}. A description of generalized EM can be found in, for example, Neal et al., “A new view of the EM algorithm that justifies incremental and other variants,” in the collection “Learning in graphical models,” (M. Jordan, editor), pages 355-368. MIT Press, 1999.

The EM algorithm consists of two steps, E and M, which are interleaved in an iterative fashion until convergence. The essential step is the expectation (E) step. This step is also known as the inference problem. The inference problem was considered in the two preferred embodiments of the method.

Given the sufficient statistics from the inference phase, it is easy to obtain parameter update equations for the maximization (M) step of the EM algorithm. Updated values of the model parameters are easily shown to be ${\hat{A}}_{i} = {\left( {\underset{t = 1}{\sum\limits^{T - 1}}{\langle{x_{t}x_{t - 1}^{\prime}{s_{t}(i)}}\rangle}} \right)\left( {\underset{t = 1}{\sum\limits^{T - 1}}{\langle{x_{t - 1}x_{t - 1}^{\prime}{s_{t}(i)}}\rangle}} \right)^{- 1}}$ ${\hat{Q}}_{i} = {\left( {{\underset{t = 1}{\sum\limits^{T - 1}}{\langle{x_{t}x_{t}^{\prime}{s_{t}(i)}}\rangle}} - {{\hat{A}}_{i}{\langle{x_{t - 1}x_{t}^{\prime}{s_{t}(i)}}\rangle}}} \right)\left( {\underset{t = 1}{\sum\limits^{T - 1}}{\langle{s_{t}(i)}\rangle}} \right)^{- 1}}$ $\hat{C} = {\left( {\underset{t = 0}{\sum\limits^{T - 1}}{y_{t}{\langle x_{t}^{\prime}\rangle}}} \right)\left( {\underset{t = 0}{\sum\limits^{T - 1}}{\langle{x_{t}x^{\prime}}\rangle}} \right)^{- 1}}$ $\hat{R} = {\frac{1}{T}{\underset{t = 0}{\sum\limits^{T - 1}}\left( {{y_{t}y_{t}^{\prime}} - {\hat{C}{\langle x_{t}\rangle}y_{t}^{\prime}}} \right)}}$ $\hat{\Pi} = {\left( {\underset{t = 1}{\sum\limits^{T - 1}}{\langle{s_{t}s_{t - 1}^{\prime}}\rangle}} \right){{diag}\left( {\underset{t = 1}{\sum\limits^{T - 1}}{\langle s_{t}\rangle}} \right)}^{- 1}}$ π̂₀ = ⟨s₀⟩.

The operator <*> denotes conditional expectation with respect to the posterior distribution, e.g. <x_(t)>=Σ_(s)∫_(x)x_(t)P(X,S|Y).

All the variable statistics are evaluated before updating any parameters. Notice that the above equations represent a generalization of the parameter update equations of classical (non-switching) LDS models.

The coupled E and M steps are the key components of the SLDS learning method. While the M step is the same for all preferred embodiments of the method, the E-step will vary with the approximate inference method.

APPLICATIONS OF SLDS

We applied our SLDS framework to the analysis of two categories of fronto-parallel motion: walking and jogging. Fronto-parallel motions exhibit interesting dynamics and are free from the difficulties of 3-D reconstruction. Experiments can be conducted easily using a single video source, while self-occlusions and cluttered backgrounds make the tracking problem non-trivial.

The kinematics of the figure are represented by a 2-D Scaled Prismatic Model (SPM). This model is described in Morris et al., “Singularity analysis for articulated object tracking,” Proceedings of Computer Vision and Pattern Recognition, pages 289-296, Santa Barbara, Calif., Jun. 23-25, 1998. The SPM lies in the image plane, with each link having one degree of freedom (DOF) in rotation and another DOF in length. A chain of SPM transforms can model the image displacement and foreshortening effects produced by 3-D rigid links. The appearance of each link in the image is described by a template of pixels which is manually initialized and deformed by the link's DOFs.

In our figure tracking experiments, we analyzed the motion of the legs, torso, and head, while ignoring the arms. Our kinematic model had eight DOFs, corresponding to rotations at the knees, hip and neck.

Learning

The first task we addressed was learning an SLDS model for walking and running. The training set consisted of eighteen sequences of six individuals jogging and walking at a moderate pace. Each sequence was approximately fifty frames in duration. The training data consisted of the joint angle states of the SPM in each image frame, which were obtained manually.

The two motion types were each modeled as multi-state SLDSs and then combined into a single complex SLDS. The measurement matrix in all cases was assumed to be identity, C=I. Initial state segmentation within each motion type was obtained using unsupervised clustering in a state space of some simple dynamics model, e.g., a constant velocity model. Parameters of the model (A, Q, R, x₀, P, π₀) were then reestimated using the EM-learning framework with approximate Viterbi inference. This yielded refined segmentation of switching states within each of the models.

FIG. 9 illustrates learned segmentation of a “jog” motion sequence. A two-state SLDS model was learned from a set of exemplary “jog” motion measurement sequences, an example of which is shown in the bottom graph. The top graph 62 depicts decoded switching states (<s_(t)>) inferred from the measurement sequence y_(t), shown in the bottom graph 64, using the learned “jog” SLDS model.

Classification

The task of classification is to segment a state space trajectory into a sequence of motion regimes, according to learned models. For instance, in gesture recognition, one can automatically label portions of a long hand trajectory as some predefined, meaningful gestures. The SLDS framework is particularly suitable for motion trajectory classification.

FIG. 10 illustrates the classification of state space trajectories. Learned SLDS model parameters are used in approximate Viterbi inference 74 to decode a best sequence 78 of models 76 corresponding to the state space trajectory 70 to be classified.

The classification framework follows directly from the framework for approximate Viterbi inference in SLDSs, described previously. Namely, the approximate Viterbi inference 74 yields a sequence of switching states S_(T) (regime indexes) 70 that best describes the observed state space trajectory 70, assuming some current SLDS model parameters. If those model parameters are learned on a corpus of representative motion data, applying the approximate Viterbi inference on a state trajectory from the same family of motions will then result in its segmentation into the learned motion regimes.

Additional “constraints” 72 can be imposed on classification. Such constraints 72 can model expert knowledge about the domain of trajectories which are to be classified, such as domain grammars, which may not have been present during SLDS model learning. For instance, we may know that out of N available SLDS motion models, only M<N are present in the unclassified data. We may also know that those M models can only occur in some particular, e.g., deterministically or stochastically, known order. These classification constraints can then be superimposed on the SLDS motion model parameters in the approximate Viterbi inference to force the classification to adhere to them. Hence, a number of natural language modeling techniques from human speech recognition, for example, as discussed in Jelinek, “Statistical methods for speech recognition,” MIT Press, 1998, can be mapped directly into the classification SLDS domain.

Classification can also be performed using variational and GPB2 inference techniques. Unlike with Viterbi inference, these embodiments yield a “soft”classification, i.e., each switching state at every time instance can be active with some potentially non-zero probability. A soft switching state at time t is $\underset{i = 0}{\sum\limits^{S - 1}}{i{{\langle{s_{t}(i)}\rangle}.}}$

To explore the classification ability of our learned model, we next considered segmentation of sequences of complex motion, i.e., motion consisting of alternations of “jog” and “walk.” Test sequences were constructed by concatenating in random order randomly selected and noise-corrupted training sequences. Transitions between sequences were smoothed using B-spline smoothing. Identification of two motion “regimes” was conducted using the proposed framework in addition to a simple HMM-based segmentation. Multiple choices of linear dynamic system and “jog”/“walk” switching orders were also compared.

Because “jog” and “walk” SLDS models were learned individually, whereas the test data contained mixed “jog” and “walk” regimes, it was necessary to impose classification constraints to combine the two models into a single “jog+walk” model. Since no preference was known a priori for either of the two regimes, the two were assumed equally likely but complementary, and their individual two-state switching models P_(jog) and P_(walk) were concatenated into a single four-state switching model P_(jog+walk). Each state of this new model simply carried over the LDS parameters of the individual “jog” and “walk” models.

Estimates of “best” switching states <s_(t)> indicated which of the two models can be considered to be driving the corresponding motion segment.

FIG. 11 illustrates an example of segmentation, depicting true switching state sequence (sequence of jog-walk motions) in the top graph, followed by HMM, Viterbi, GPB2, and variational state estimates using one switching state per motion type models, first order SLDS model.

FIG. 11 contains several graphs which illustrate the impact of different learned models and inference methods on classification of “jog” and “walk” motion sequences, where learned “jog” and “walk” motion models have two switching states each. Continuous system states of the SLDS model contain information about angle of the human figure joints. The top graph 80 depicts correct classification of a motion sequence containing “jog” and “walk” motions (measurement sequence is not shown). The remaining graphs 82-88 show inferred classifications using, respectively from top to bottom, SLDS model with Viterbi inference, SLDS model with GPB2 inference, SLDS model with variational inference, and HMM model.

Similar results were obtained with four switching states each and second order SLDS.

Classification experiments on a set of 20 test sequences gave an error rate of 2.9% over a total of 8312 classified data points, where classification error was defined as the difference between inferred segmentation (the estimated switching states) and true segmentation (true switching states) accumulated over all sequences, ${error} = {\underset{t = 0}{\sum\limits^{T - 1}}{{{{\langle s_{t}\rangle} - s_{{true},t}}}.}}$

Tracking

The SLDS learning framework can improve the tracking of target objects with complex dynamical behavior, responsive to a sequence of measurements. A particular embodiment of the present invention tracks the motion of the human figure, using frames from a video sequence. Tracking systems generally have three identifiable steps: state prediction, measurement processing and posterior update. These steps are exemplified by the well-known Kalman Filter, described in Anderson and Moore, “Optimal Filtering,” Prentice Hall, 1979.

FIG. 12 is a standard block diagram for a Kalman Filter. A state prediction module 500 takes as input the state estimate from the previous time instant, {circumflex over (x)}_(t−1|t−1). The output of the prediction module 500 is the predicted state {circumflex over (x)}_(t|t−1). The measurement processing module 501 takes the predicted state, generates a corresponding predicted measurement, and combines it with the actual measurement y_(t) to form the “innovation” z_(t). For an image, for example, states might be parameters such as angles, lengths and positions of objects or features, while the measurements might be the actual pixels. The predicted measurements might then be the predictions of pixel values and/or pixel locations. The innovation z_(t) measures the difference between the predicted and actual measurement data.

The innovation z_(t) is passed to the posterior update module 502 along with the prediction. They are fused using the Kalman gain matrix to form the posterior estimate {circumflex over (x)}_(t|t).

The delay element 503 indicates the temporal nature of the tracking process. It models the fact that the posterior estimate for the current frame becomes the input for the filter in the next frame.

FIG. 13 illustrates the operation of the prediction module 500 for the specific case of figure tracking. The dashed line 510 shows the position of a skeletal representation of the human figure at time t−1. The predicted position of the figure at time t is shown as a solid line 511. The actual position of the figure in the video frame at time t is shown as a dotted line 512. The unknown state x_(t) represents the actual skeletal position for the sketched figure.

We now describe the process of computing the innovation z_(t) which is the task of the measurement processing module 501, in the specific case of figure tracking using template features. A template is a region of pixels, often rectangular in shape. Tracking using template features is described in detail in James M. Rehg and Andrew P. Witkin, “Visual Tracking with Deformation Models,” Proceedings of IEEE Conference on Robotics and Automation, April 1991, Sacramento Calif., pages 844-850. In the figure tracking application, a pixel template is associated with each part of the figure model, describing its image appearance. The pixel template is an example of an “image feature model” which describes the measurements provided by the image.

For example, two templates can be used to describe the right arm, one for the lower and one for the upper arm. These templates, along with those for the left arm and the torso, describe the appearance of the subject's shirt. These templates can be initialized, for example, by capturing pixels from the first frame of the video sequence in which each part is visible. The use of pixel templates in figure tracking is described in detail in Daniel D. Morris and James M. Rehg, “Singularity Analysis for Articulated Object Tracking,” Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, June 1998, Santa Barbara Calif., pages 289-296.

Given a set of initialized figure templates, the innovation is the difference between the template pixel values and the pixel values in the input video frame which correspond to the predicted template position. We can define the innovation function that gives the pixel difference as a function of the state vector x and pixel index k:

z(x,k)=I _(t)(pos(x,k))−T(k)  (Eq. 17)

We can then write the innovation at time t as

z _(t)(k)=z({circumflex over (x)}_(t|t−1) ,k)  (Eq. 18)

Equation 18 defines a vector of pixel differences, z_(t), formed by subtracting pixels in the template model T from a region of pixels in the input video frame I_(t) under the action of the figure state. In order to represent images as vectors, we require that each template pixel in the figure model be assigned a unique index k. This can be easily accomplished by examining the templates in a fixed order and scanning the pixels in each template in raster order. Thus z_(t)(k) represents the innovation for the k th template pixel, whose intensity value is given by T(k). Note that the contents of the template T(k) could be, for example, gray scale pixel values, color pixel values in RGB or YUV, Laplacian filtered pixels, or in general any function applied to a region of pixels. We use z(x) to denote the vector of pixel differences described by Equation (e1). Note that z_(t) does not define an innovation process in the strict sense of the Kalman filter, since the underlying system is nonlinear and non-Gaussian.

The deformation function pos(x,k) in Equation 17 gives the position of the k th template pixel, with respect to the input video frame, as a function of the model state x. Given a state prediction {circumflex over (x)}_(t|t−1), the transform pos({circumflex over (x)}_(t|t−1),*) maps the template pixels into the current image, thereby selecting a subset of pixel measurements. In the case of figure tracking, the pos( ) function models the kinematics of the figure, which determine the relative motion of its parts.

FIG. 14 illustrates two templates 513 and 514 which model the right arm 525. Pixels that make up these templates 513, 514 are ordered, with pixels T₁ through T₁₀₀ belonging to the upper arm template 513, and pixels T₁₀₁ through T₂₀₀ belonging to the lower arm template 514. The mapping pos(x,k) is also illustrated. The location of each template 513, 514 under the transformation is shown as 513A, 514A respectively. The specific transformations of two representative pixel locations, T₄₈ and T₁₈₁, are also illustrated.

Alternatively, the boundary of the figure in the image can be expressed as a collection of image contours whose shapes can be modeled for each part of the figure model. These contours can be measured in an input video frame and the innovation expressed as the distance in the image plane between the positions of predicted and measured contour locations. The use of contour features for tracking is described in detail in Demetri Terzopoulos and Richard Szeliski, “Tracking with Kalman Snakes,” which appears in “Active Vision,” edited by Andrew Blake and Alan Yuille, MIT Press, 1992, pages 3-20. In general, the tracking approach outlined above applies to any set of features which can be computed from a sequence of measurements.

In general, each frame in an image sequence generates a set of “image feature measurements,” such as pixel values, intensity gradients, or edges. Tracking proceeds by fitting an “image feature model,” such as a set of templates or contours, to the set of measurements in each frame.

A primary difficulty in visual tracking is the fact that the image feature measurements are related to the figure state only in a highly nonlinear way. A change in the state of the figure, such as raising an arm, induces a complex change in the pixel values produced in an image. The nonlinear function I(pos(x,*)) models this effect. The standard approach to addressing this nonlinearity is to use the Iterated Extended Kalman Filter (IEKF), which is described in Anderson and Moore, section 8.2. In this approach, the nonlinear measurement model in Equation 17 is linearized around the state prediction {circumflex over (x)}_(t|t−1).

The linearizing function can be defined as $\begin{matrix} {{M_{t}\left( {x,k} \right)} = {\bigtriangledown \quad {I_{t}\left( {{pos}\left( {x,k} \right)} \right)}^{\prime}\quad \frac{\partial{pos}}{\partial x}\left( {x,k} \right)}} & \text{(Eq.~~19)} \end{matrix}$

The first term on the right in Equation 19, ∇I_(t)(pos(x,k))′, is the image gradient ∇I_(t), evaluated at the image position of template pixel k. The second term, ${\frac{\partial{pos}}{\partial x}\left( {x,k} \right)},$

is a 2×N kinematic Jacobian matrix, where N is the number of states, or elements, in x. Each column of this Jacobian gives the displacement in the image at pixel position k due to a small change in one of the model states. M_(t)(x,k) is a 1×N row vector. We denote by M_(t)(x) the M×N matrix formed by stacking up these rows for each of the M pixels in the template model. The linearized measurement model can then be written:

{overscore (C)} _(t) =M _(t)({circumflex over (x)} _(t|t−1))  (Eq. 20)

The standard Kalman filter equations can then be applied using the linearized measurement model from Equation 20 and the innovation from Equation 18. In particular, the posterior update equation (analogous to Equation 11 in the linear case) is:

{circumflex over (x)} _(t|t) ={circumflex over (x)} _(t|t−1) +L _(t) z _(t)  (Eq. 21)

where L_(t) is the appropriate Kalman gain matrix formed using {overscore (C)}_(t).

FIG. 15 illustrates a block diagram of the IEKF. In comparison to FIG. 12, the measurement processing 501 and posterior update 502 blocks within the dashed box 506 are now iterated P times with the same measurement y_(t), for some predetermined number P, before the final output 507 is reported and a new measurement y_(t) is introduced. This iteration produces a series of innovations z″_(t) and posterior estimates {circumflex over (x)}_(t|t) ^(n) which result from successive linearizations of the measurement model around the previous posterior estimate {circumflex over (x)}_(t|t) ^(n−1). The iterations are initialized by setting {circumflex over (x)}_(t|t) ¹={circumflex over (x)}_(t|t−1). Upon exiting, we set {circumflex over (x)}_(t|t)={circumflex over (x)}_(t|t) ^(P).

In cases where the prediction is far from the correct state estimate, these iterations can improve accuracy. We denote as the “IEKF update module” 506 the subsystem of measurement processing 501 and posterior update 502 blocks, as well as delay 504, although in practice, delays 503 and 504 may be the same piece of hardware and/or software.

Clearly the quality of the IEKF solution depends upon the quality of the state prediction. The linearized approximation is only valid within a small range of state values around the current operating point, which is initialized by the prediction. More generally, there are likely to be many background regions in a given video frame that are similar in appearance to the figure. For example, a template that describes the appearance of a figure wearing a dark suit might match quite well to a shadow cast against a wall in the background. The innovation function defined in Equation 17 can be viewed as an objective function to be minimized with respect to x. It is clear that this function will in general have many local minima. The presence of local minima poses problems for any iterative estimation technique such as IEKF. An accurate prediction can reduce the likelihood of becoming trapped in a local minima by bringing the starting point for the iterations closer to the correct answer.

As alluded to previously, the dynamics of a complex moving object such as the human figure cannot be expressed using a simple linear dynamic model. Therefore, it is unlikely that the standard IEKF framework described above would be effective for figure tracking in video. A simple linear prediction of the figure's state would not provide sufficient accuracy in predicting figure motion.

The SLDS framework of the present invention, in contrast, can describe complex dynamics as a switched sequence of linear models. This framework can form an accurate predictor for video tracking. A straightforward implementation of this approach can be extrapolated directly from the approximate Viterbi inference.

In the first tracking embodiment, the Viterbi approach generates a set of S² hypotheses, corresponding to all possible transitions i,j between linear models from time step t−1 to t. Each of these hypotheses represents a prediction for the continuous state of the figure which selects a corresponding set of pixel measurements. It follows that each hypothesis has a distinct innovation equation:

z _(t|t−1,i,j) =z({circumflex over (x)} _(t|t−1,i,j))  (Eq. 22)

defined by Equation 17. Note that the only difference in comparison to Equation 18 is the use of the SLDS prediction in mapping templates into the image plane.

This embodiment, using SLDS models, is an application of the Viterbi inference algorithm described earlier to the particular measurement models used in visual tracking.

FIG. 16 is a block diagram which illustrates this approach. The Viterbi prediction block 510 generates the set of S² predictions {circumflex over (x)}_(t|t−1,i,j) according to Equation 9.

A selector 511 takes these inputs and selects, for each of the S possible switching states at time t, the most likely previous state ψ_(t−1,i), as defined in Equation 13. This step is identical to the Viterbi inference procedure described earlier, except that the likelihood of the measurement sequence is computed using image features such as templates or contours. More specifically, Equation 7 expresses the transition likelihood J_(t|t−1,i,j) as the product of the measurement probability and the transition probability from the Markov chain.

In the case of template features, the measurement probability can be written $\begin{matrix} \begin{matrix} {{P\left( y_{{t|{t - 1}},i,j} \right)} = \quad {P\left( {{\left. y_{t} \middle| s_{t} \right. = e_{i}},{s_{t - 1} = e_{j}},{S_{t - 2}^{*}(j)}} \right)}} \\ {\approx \quad {N\left( {{z_{{t|{t - 1}},i,j};0},{{{\overset{\_}{C}}_{{t|{t - 1}},i,j}\Sigma_{{t|{t - 1}},i,j}{\overset{\_}{C}}_{{t|{t - 1}},i,j}^{\prime}} + R}} \right)}} \end{matrix} & \text{(Eq.~~23)} \end{matrix}$

where {overscore (C)}_(t|t−1,i,j)=M_(t)({circumflex over (x)}_(t|t−1,i,j)). The difference between Equations 23 and 12 is the use of the linearized measurement model for template features. By changing the measurement probability appropriately, the SLDS framework above can be adopted to any tracking problem.

The output of the selector 511 is a set of S hypotheses corresponding to the most probable state predictions given the measurement data. These are input to the IEKF update block 506, which filters these predictions against the measurements to obtain a set of S posterior state estimates. This block was illustrated in FIG. 15. This step applies the standard equations for the IEKF to features such as templates or contours, as described in Equations 17 through 21. The posterior estimates can be decoded and smoothed analogously to the case of Viterbi inference.

In computing the measurement probabilities, the selector 511 must make S² comparisons between all of the model pixels and the input image. Depending upon the size of the target and the image, this may represent a large computation. This computation can be reduced by considering only the Markov process probabilities and not the measurement probabilities in computing the best S switching hypotheses. In this case, the best hypotheses are given by

i _(t)*=arg min_(j){−log Π(i,j)+J _(t−1,j)}  (Eq. 24)

This is a substantial reduction in computation, at the cost of a potential loss of accuracy in picking the best hypotheses.

FIG. 17 illustrates a second tracking embodiment, in which the order of the IEKF update module 506 and selector 511A is reversed. In this case, the set of S² predictions are passed directly to the IEKF update module 506. The output of the update module 506 is a set of S² filtered estimates {circumflex over (x)}_(t|t,i,j), each of which is the result of P iterations of the IEKF. As with the embodiment of FIG. 16, it is still necessary to reduce the total set of estimates to S hypotheses in order to control the complexity of the tracker.

The selector 511A chooses the most likely transition j→i for each switching state based on the filtered estimates. This is analogous to the Viterbi inference case, which was described in the embodiment of FIG. 16 above. The key step is to compute the switching costs J_(t,i) defined in Equations 5 and 6. The difference in this case comes from the fact that the probability of the measurement is computed using the filtered posterior estimate rather than the prediction. This probability is given by:

P(y _(t|t,i,j))=P(y _(t) |s _(t) =e _(i) ,s _(t−1) =e _(j) ,{circumflex over (x)} _(t|t,i,j) ,S _(t−2)*(j))≈N(z({circumflex over (x)} _(t|t,i,j));0,M _(t)({circumflex over (x)} _(t|t,i,j))Σ_(t|t,i,j) M _(t)({circumflex over (x)} _(t|t,i,j))′+R)

The switching costs are then given by

J _(t,i)=max_(j) {J _(t|t,i,j) J _(t−1,j)}  (Eq. 25A)

where the “posterior transition probability” from state j at time t−1 to state i at time t is written

J_(t|t,i,j) =P(y _(t|t,i,j))P(s _(t) =e _(i) |s _(t−1) =e _(j))  (Eq. 25B)

The selector 511A selects the transition j*→i, where j*=arg max_(j){J_(t|t,i,j),J_(t−1|j)}. The state j* is the “optimal posterior switching state,” and its value depends upon the posterior estimate for the continuous state at time t.

The potential advantage of this approach is that all of the S² predictions are given an opportunity to filter the measurement data. Since the system is nonlinear, it is possible that a hypothesis which has a low probability following prediction could in fact produce a posterior estimate which has a high probability.

The set of S² predictions used in the embodiments of FIGS. 16 and 17 can be viewed as specifying a set of starting points for local optimization of an objective function defined by ∥z_(t)(x)∥². In fact, as the covariance for the plant noise approaches infinity, the behavior of the IEKF module 506 approaches steepest-descent gradient search because the dynamic model no longer influences the posterior estimate.

When the objective function has many local minima relative to the uncertainty in the state dynamics, it may be advantageous to use additional starting points for search beyond the S² values provided by the SLDS predictor. This can be accomplished, for example, by drawing additional starting points at random from a “continuous state sampling density.” Additional starting points can increase the chance that one of the IEKF tracks will find the global minimum. We therefore first describe a procedure for generating additional starting points and then describe how they can be incorporated into the previous two embodiments.

A wide range of sampling procedures is available for generating additional starting points for search. In order to apply the IEKF update module 506 at a given starting point, a state prediction must be specified and a particular dynamic model selected. The easiest way to do this is to assume that new starting points are going to be obtained by sampling from the mixture density defined by the set of S² predictions. This density can be derived as follows: $\begin{matrix} {{P\left( x_{t} \middle| Y_{t - 1} \right)} = \quad {\sum\limits_{i,j}\quad {P\left( {{\left. x_{t} \middle| s_{t} \right. = e_{i}},{s_{t - 1} = e_{j}},S_{t - 2},Y_{t - 1}} \right)}}} \\ {\quad {P\left( {{s_{t} = e_{i}},{s_{t - 1} = e_{j}},S_{t - 2},Y_{t - 1}} \right)}} \\ {\approx \quad {\sum\limits_{i,j}\quad {P\left( {{\left. x_{t} \middle| s_{t} \right. = e_{i}},{s_{t - 1} = e_{j}},{S_{t - 2} = {S_{t - 2}^{*}(j)}},Y_{t - 1}} \right)}}} \\ {\quad {P\left( {{s_{t} = e_{i}},{s_{t - 1} = e_{j}},{S_{t - 2} = \left. {S_{t - 2}^{*}(j)} \middle| Y_{t - 1} \right.}} \right)}} \end{matrix}$

The first term in the summed product is a Gaussian prior density from the Viterbi inference step. The second term is its likelihood, a scalar probability which we denote α_(i,j). Rewriting in the form of a mixture of S² Gaussian kernels, i.e., the predictions, yields: $\begin{matrix} \begin{matrix} {{P\left( x_{t} \middle| Y_{t - 1} \right)} \approx \quad {\sum\limits_{i,j}{\alpha_{i,j}{K_{i,j}\left( x_{t} \right)}}}} \\ {= \quad {\sum\limits_{i,j}{\alpha_{i,j}{N\left\lbrack {{x_{t};{\hat{x}}_{{t|{t - 1}},i,j}},\Sigma_{{t|{t - 1}},i,j}} \right\rbrack}}}} \end{matrix} & \left( {{Eq}.\quad 26} \right) \end{matrix}$

where α_(i,j) can be further expanded as $\begin{matrix} {\alpha_{i,j} = \quad {{p\left( {{s_{t} = {\left. e_{i} \middle| s_{t - 1} \right. = e_{j}}},{S_{t - 2}^{*}(j)},Y_{t - 1}} \right)}{P\left( {{s_{t - 1} = e_{j}},\left. {S_{t - 2}^{*}(j)} \middle| \left. Y_{t - 1} \right) \right.} \right.}}} \\ {= \quad {{P\left( {s_{t} = {\left. e_{i} \middle| s_{t - 1} \right. = e_{j}}} \right)}\quad \frac{P\left( {{s_{t - 1} = e_{j}},{S_{t - 2}^{*}(j)},Y_{t - 1}} \right)}{P\left( Y_{t - 1} \right)}}} \end{matrix}$

Applying Equation 1 to the first term and Equations 5 and 8 to the second term yields $\begin{matrix} {\alpha_{i,j} = \frac{{\Pi \left( {i,j} \right)}J_{{t - 1},j}}{P\left( Y_{t - 1} \right)}} & \text{(Eq.~~27)} \end{matrix}$

All of the terms in the numerator of Equation 27 are directly available from the Viterbi inference method. The denominator is a constant normalizing factor which can be rewritten to yield $\begin{matrix} {\alpha_{i,j} = \frac{{\Pi \left( {i,j} \right)}J_{{t - 1},j}}{\sum\limits_{i,j}{{\Pi \left( {i,j} \right)}J_{{t - 1},j}}}} & \text{(Eq.~~28)} \end{matrix}$

Equations 26 and 28 define a “Viterbi mixture density” from which additional starting points for search can be drawn. The steps for drawing R additional points are as follows:

Find mixture parameters, α_(i,j),x_(t|t−1,i,j), and Σ_(t|t−1,i,j) from Viterbi prediction

For r=1 to R

Select a kernel K_(i) _(r) _(,j) _(r) at random according to the discrete distribution {α_(i,j)}

Select a state sample {overscore (x)}_(i) _(r) _(,j) _(r) at random according to the predicted Gaussian distribution K_(i) _(r) _(,j) _(r)

End

The rth sample is associated with a specific prediction (i_(r),j_(r)), making it easy to apply the IEKF.

Given a set of starting points, we can apply the IEKF approach by modifying the IEKF equations to support linearization around arbitrary points. Consider a starting point {overscore (x)}. The nonlinear measurement model for template tracking can be written

z _(t)(x _(t))=w _(t)

Expanding around {overscore (x)}, discarding high-order terms and rearranging gives

{overscore (y)} _(t) =z _(t)({overscore (x)})−{overscore (C)} _(t) {overscore (x)}=−{overscore (C)} _(t) x _(t) +w _(t)

where the auxiliary measurement {overscore (y)}_(t) has been defined to give a measurement model in standard form. It follows that the posterior update equation is

{circumflex over (x)} _(t|t) ={circumflex over (x)} _(t|t−1) +L _(t)(z _(t)({overscore (x)})+{overscore (C)} _(t)({circumflex over (x)} _(t|t−1) −{overscore (x)}))  (Eq. 29)

Note that if {overscore (x)}=x_(t|t−1), which is the standard operating point for IEKF, then the above reduces to Equation 21. The additional term captures the effect of the new operating point. Equation 29 defines a modification of the IEKF update block to handle arbitrary starting points in computing the posterior update.

Note that the smoothness of the underlying nonlinear measurement model will determine the region in state space over which the linearized model {overscore (C)}_(t) is an accurate approximation. This region must be large enough relative to the distance ∥{circumflex over (x)}_(t|t−1)−{overscore (x)}∥. This requirement may not be met by a complex objective function.

An alternative in that case is to apply a standard gradient descent approach instead of IEKF, effectively discounting the role of the prior state prediction. A multiple hypothesis tracking (MHT) approach which implements this procedure is described in Tat-Jen Cham and James M. Rehg, “A Multiple Hypothesis Approach to Figure Tracking,” Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, June 1999, Ft. Collins, Colo., pages 239-245, incorporated herein by reference in its entirety. Although this article does not describe tracking with an SLDS model or sampling from an SLDS prediction, the MHT procedure provides a means for propagating a set of sampled states given a complex measurement function. This means can be used as an alternative to the IEKF update step.

FIG. 18 is a block diagram of a tracking embodiment which combines SLDS prediction with sampling from the prior mixture density to perform tracking. The output of the Viterbi predictor 510 follows two paths. The top path is similar to the embodiment of FIG. 16. The predictions are processed by a selector 515 and then input to an IEKF update block 506.

Along the bottom path in FIG. 18, the predictions are input to a sample generator 517, which produces a new set of R sample points for filtering, {{overscore (x)}_(i) _(r) _(,j) _(r) }_(r=1) ^(R). This set is unioned with the SLDS predictions from the selector 515 and input to the IEKF block 506. The output of the IEKF block 506 is a joint set of filtered estimates, corresponding to the SLDS predictions, which we now write as {overscore (x)}_(t|t,i), and the sampled states {overscore (x)}_(t|t,i) _(r) _(,j) _(r) .

This combined output forms the input to a final selector 519 which selects one filtered estimate for each of the switching states to make up the final output set. This selection process is identical to the ones described earlier with respect to FIG. 17, except that there now can be more than one possible estimate for a given state transition (i,j), corresponding to different starting points for search.

FIG. 19 illustrates yet another tracking embodiment. The output of Viterbi prediction, which comprises “Viterbi estimates,” is input directly to an IEKF update block 506 while the output of the sample generator 517 goes to an MHT block 520, which implements the method of Cham and Rehg referred to above. As with the embodiment of FIG. 18, these two sets of filtered estimates are passed to a final selector 522. This final selector 522 chooses S of the posterior estimates, one for each possible switching state, as the final output. As with the embodiment of FIG. 18, this selector uses the posterior switching costs defined in Equation 25B.

It should be clear from the embodiments of FIGS. 18 and 19 that other variations on the same basic approach are also possible. For example, an additional selector can be added following the predictor of FIG. 19, or the MHT block can be replaced by a second IEKF block.

It has been assumed that the selector would be selecting a single most likely state distribution from a set of distributions associated with a particular switching state. Another possibility is to find a single distribution which most closely matches all of the possibilities. This is the Generalized Psuedo Bayesian approximation GPB2, described earlier. In any of the selector blocks for the embodiments discussed above, GPB2 could be used in place of Viterbi approximation as a method for reducing multiple hypotheses for a particular switching state down to a single hypothesis. In cases where no single distribution contains a majority of the probability mass for the set of hypotheses, this approach may be advantageous. The application of the previously described process for GPB2 inference to these embodiments is straightforward.

Synthesis and Interpolation

SLDS was introduced as a “generative” model; trajectories in the state space can be easily generated by simulating an SLDS. Nevertheless, SLDSs are still more commonly employed as a classifier or a filter/predictor than as a generative model. We now formulate a framework for using SLDSs as synthesizers and interpolators.

Consider again the generative model described previously. Provided SLDS parameters have been learned from a corpus of motion trajectories, driving the generative SLDS model with the appropriate state and measurement noise processes and switching model, will yield a state space trajectory consistent with that corpus. In other words, one will draw a sample (trajectory) defined by the probability distribution of the SLDS.

FIG. 20 illustrates a framework for synthesis of state space trajectories in which the SLDS is used as a generative model within a synthesis module 410. Given the parameters of the SLDS model 411, obtained using either SLDS learning or some other techniques, a switching state sequence s_(t) is first synthesized in the switching state synthesis module 412 by sampling from a Markov chain with state transition probability matrix Π and initial state distribution π₀. The continuous state sequence x_(t) is then synthesized in the continuous state synthesis module 413 by sampling from a LDS with parameters A(s_(t)), Q(s_(t)), C, R, x₀(s₀), and Q₀(s₀).

The above procedure will produce a random sequence of samples from the SLDS model. If an average noiseless state trajectory is desired, the synthesis can be run with LDS noise parameters (Q(s_(t)), R, Q₀(s₀)) set to zero and switching states whose duration is equal to average state durations, as determined by the switching state transition matrix Π. For example, this would result in sequences of prototypical walk or jog motions, whereas the random sampling would exhibit deviations from such prototypes. Intermediate levels of randomness can be achieved by scaling the SLDS model noise parameters with factors between 0 and 1.

The model parameters can also be modified to meet new constraints that were, for instance, not present in the data used to learn the SLDS. For example, initial state mean x₀, variance Q₀ and switching state distribution π can be changed so as to force the SLDS to start in some arbitrary state x_(a) of regime i_(a), e.g., to start simulation in a “walking” regime i_(a) with figure posture x_(a). To achieve this, set x₀(i_(a))=x_(a), Q₀(i_(a))=0, π(i_(a))=1, and then proceed with the synthesis of this constrained model.

A framework of optimal control can be used to formalize synthesis under constraints. Optimal control of linear dynamic systems is described, for example, in B. Anderson and J. Moore, “Optimal Control: Linear Quadratic Methods,” Prentice Hall, Englewood Cliffs, N.J., 1990. For a LDS, the optimal control framework provides a way to design an optimal input or control u_(t) to the LDS, such that the LDS state x_(t) gets as close as possible to a desired state x_(t) ^(d). The desired states can also be viewed as constraint points. The same formalism can be applied to SLDSs. Namely, the system equation, Equation 1, can be modified as

x _(t+1) =A(s _(t+1))x _(t) +u _(t+1) v _(t+1)(s _(t+1))  (Eq. 1A)

FIG. 21 is a dependency graph of the modified SLDS 550 with added controls u_(t) 552. A goal is to find u_(t) that makes x_(t) as close as possible to a given x_(t) ^(d). Usually, a quadratic measure of closeness is used, i.e., a control u_(t) is desired such that the cost V^((x)), or value function $V^{(x)} = {\langle{\overset{T - 1}{\sum\limits_{t = 0}}\left\lbrack {{\left( {x_{t} - x_{t}^{d}} \right)^{\prime}{W_{t}^{(x)}\left( {x_{t} - x_{t}^{d}} \right)}} + {u_{t}^{\prime}W_{t}^{(u)}u_{t}}} \right\rbrack}\rangle}$

is minimized, where W_(t) ^((x)) and W_(t) ^((u)) are weight matrices. The optimal control is then ${\hat{u}}_{t} = {\arg \quad {\min\limits_{u}V^{(x)}}}$

For instance, if a SLDS is used to simulate a motion of the human figure, x_(t) ^(d) might correspond to a desired figure posture at key frame t, and W_(t) ^((x)) might designate the key frame, i.e., W_(t) ^((x)) is large for the key frame, and small otherwise.

In addition to “closeness” constraints, other types of constraints can similarly be added. For example, one can consider a minimum-time constraint where the terminal timer T is to be minimized with the optimal control û_(t). In that case, the value function to be minimized becomes V^((x))←V^((x))+T.

Other types of constraints that can be cast in this framework are the inequality or bounding constraints on the state x_(t) or control u_(t) (e.g., x_(t)>x^(min), u_(t<u) ^(max)). Such constraints could prevent, for example, the limbs of a simulated human figure from assuming physically unrealistic postures, or the control u_(t) from becoming unrealistically large.

If s_(t) is known, the solution for the best control u_(t) can be derived using the framework of linear quadratic regulators (LQR) for time-varying LDS. When s_(t) is not known, the estimates of s_(t) obtained from Viterbi or variational inference can be used and the best control solution can then be found using LQR.

Alternatively, as shown in FIG. 22, the SLDS 560 can be modified to include inputs 562, i.e., controls, to the switching state. The modified SLDS 560 is then described by the following equation:

Pr(s _(t+1) =i|s _(t) =j,a _(t+1) =k)=Π^((a))(i,j,k)

where a_(t) represents the control of the switching state at time t, and Π^((a))(i,j,k) defines a conditioned switching state transition matrix which depends on the control a_(t) 562.

Using the switching control a_(t) 562, constraints imposed on the switching states can be satisfied. Such constraints can be formulated similarly to constraints for the continuous control input of FIG. 21, e.g., switching state constraints, switching input constraints and minimum-time constraints. For example, a switching state constraint can guarantee that a figure is in the walking motion regime from time t_(s) to t_(e). To find an optimal control â_(t) that satisfies those constraints, one would have to use a modified value function that includes the cost of the switching state control. A framework for the switching state optimal control could be derived from the theory of reinforcement learning and Markov decision processes. See, for example, Sutton, R. S. and Barto, A. G., “Reinforcement Learning: An Introduction,” Cambridge, Mass., MIT Press, 1998.

For example, one would like to find a_(t) which minimizes the following value function, $V^{(s)} = {- {\langle{\underset{t = 0}{\sum\limits^{T - 1}}{\gamma^{t}c_{t}\left( {s_{t},s_{t - 1},a_{t}} \right)}}\rangle}}$

Here, c_(t)(s_(t),s_(t−1),a_(t)) represents a cost for making the transition from switching state s_(t−1) to s_(t), for a given control a_(t), and γ is a discount or “forgetting” factor. The cost function c_(t) is designed to emphasize, with a low c_(t), those states that agree with imposed constraints, and to penalize, with a high c_(t), those states that violate the imposed constraints. Once the optimal control â_(t) has been designed, the modified generative SLDS model can be driven by noise to produce a synthetic trajectory.

As FIG. 23 illustrates, the SLDS system 750 can be modified to include both types of controls, continuous 572 and switching 574, as indicated by the following equation.

x _(t+1) =A(s_(t+1))x _(t) +u _(t+1) +v(s _(t+1))

Pr(s _(t+1) =i|s _(t) =j,a _(t+1) =k)=Π^((a)))(i,j,k)

In this model 570, the mixed control (u_(t),a_(t)) can lead to both a desired switching state, e.g., motion regime, and a desired continuous state, e.g., figure posture. For example, a constraint can be specified that requires the human figure to be in the walking regime i_(d) with some specific posture x_(d) at time t. As in the case of the continuous and switching state optimal controls, additional constraints such as input bounding and minimum time can also be specified for the mixed state control. To find the optimal control (û_(t),â_(t)), a value function can be used that includes the costs of the switching and the continuous state controls, e.g., V^((x))+V^((s)). Again, once the optimal control (û_(t),â_(t)) is designed, the modified generative SLDS model can be used to produce a synthetic trajectory.

FIG. 24 illustrates the framework 580, for synthesis under constraints, which utilizes optimal control. The SLDS model 582 is modified by a SLDS model modification module 584 to include the control terms or inputs. Using the modified model 585, an optimal control module 586 finds the optimal controls 587 which satisfy constraints 588. Finally, a synthesis model 590 generates synthesized trajectories 592 from the modified SLDS 585 and the optimal controls 587.

The use of optimal controls by the present invention to generate motion by sampling from SLDS models in the presence of constraints has broad applications in computer animation and computer graphics. The task of generating realistic or compelling motions for synthetic characters has long been recognized as an extremely challenging problem. One classical approach to this problem is based on the notion of spacetime constraints which was first introduced by Andrew Witkin and Michael Kass, “Spacetime Constraints,” Computer Graphics, Volume 22, Number 4, August 1988 pages 159-168. In this approach, an optimal control problem is formulated over an analytic dynamical model derived from Newtonian physics. For example, in order to make a lamp hop in a realistic way, the animator would derive the physical equations which govern the lamp's motion. These equations involve the mass distribution of the lamp and forces such as gravity that are acting upon it.

Unfortunately, this method of animation has proved to be extremely difficult to use in practice. There are two main problems. First, it is extremely difficult to specify all of the equations and parameters that are necessary to produce a desired motion. In the case of the human figure, for example, specifying all of the model parameters required for realistic motion is a daunting task. The second problem is that the resulting equations of motion are highly complex and nonlinear, and usually involve an enormous number of state variables. For example, the jumping Luxo lamp described in the Witkin and Kass paper involved 223 constraints and 394 state variables. The numerical methods which must be used to solve control problems in such a large complex state space are also difficult to work with. Their implementation is complex and their convergence to a correct solution can be problematic.

In contrast, our approach of optimal control of learned SLDS models has the potential to overcome both of these drawbacks. By using a learned switching model, desired attributes such as realism can be obtained directly from motion data. Thus, there is no need to specify the physics of human motion explicitly in order to obtain useful motion synthesis. By incorporating additional constraints through the mechanism of optimal control, we enable an animator to achieve specific artistic goals by controlling the synthesized motion. One potential advantage of our optimal control framework over the classical spacetime approach is the linearity of the SLDS model once a switching sequence has been specified. This makes it possible to use optimal control techniques such as the linear quadratic regulator which are extremely stable and well-understood. Implementations of LQR can be found in many standard software packages such as Matlab. This stands in contrast to the sequential quadratic programming methods required by the classical spacetime approach.

The spacetime approach can be extended to include the direct use of motion capture data. This is described in Michael Gleicher, “Retargetting Motion to New Characters,” Proceedings of SIGGRAPH 98, in Computer Graphics Proceedings, Annual Conference series, 1998, pages 33-42. In this method, a single sequence of human motion data is modified to achieve a specific animation objective by filtering it with a biomechanical model of human motion and adjusting the model parameters. With this method, a motion sequence of a person walking can be adapted to a figure model whose proportions are different from that of the subject. Motions can also be modified to satisfy various equality or inequality constraints.

The use of motion capture data within the spacetime framework makes it possible to achieve realistic motion without the complete specification of biomechanical model parameters. However this approach still suffers from the complexity of the underlying numerical methods. Another disadvantage of this approach in contrast to the current invention is that there is no obvious way to generate multiple examples of the same type of motion or to generate new motions by combining several examples of motion data.

In contrast, sampling repeatedly from our SLDS framework can produce motions which differ in their small details but are qualitatively consistent. This type of randomness is necessary in order to avoid awkward repetitions in an animation application. Furthermore, the learning approach makes it possible to generalize from a set of motions to new motions that have not been seen before. In contrast, the approach of retargeting is based on modifying a single instance of motion data.

Another approach to synthesizing animations from learned models is described in Matthew Brand, “Voice Puppetry,” SIGGRAPH 99 Conference Proceedings, Annual Conference Series 1999, pages 21-28 and in Matthew Brand and Aaron Hertzmann, “Style Machines,” SIGGRAPH 2000 Conference Proceedings, Annual Conference Series 2000, pages 183-192. This method uses sampling from a Hidden Markov Model to synthesize facial and figure animations learned from training data. Unlike our SLDS framework, the HMM representation is limited to using piecewise constant functions to model the feature data during learning. This can require a large number of discrete states in order to capture subtle effects when working with complex state space data.

In contrast, our framework employs a set of LDS models to describe feature data, resulting in models with much more expressive power. Furthermore, the prior art does not describe any mechanism for imposing constraints on the samples from the models. This may make it difficult to use this approach in achieving specific animation objectives.

To test the power of the learned SLDS synthesis/interpolation framework, we examined its use in synthesizing realistic-looking motion sequences and interpolating motion between missing frames. In one set of experiments, the learned walk/jog SLDS was used to generate a “synthetic walk” based on initial conditions learned by the SLDS model.

FIG. 25 illustrates a stick FIG. 220 motion sequence of the noise driven model. Depending on the amount of noise used to drive the model, the stick figure exhibits more or less “natural”—looking walk. Departure from the realistic walk becomes more evident as the simulation time progresses. This behavior is not unexpected as the SLDS in fact learns locally consistent motion patterns. FIG. 25 illustrates a synthesized walk motion over 50 frames using SLDS as a generative model. The states of the synthesized motion are shown on the graph 222.

Another realistic situation may call for filling in a small number of missing frames from a large motion sequence. SLDS can then be utilized as an interpolation function. In another set of experiments, we employed the learned walk/jog model to interpolate a walk motion over two sequences with missing frames. Missing-frame constraints were included in the interpolation framework by setting the measurement variances corresponding to those frames to infinity. The visual quality of the interpolation and the motion synthesized from it was high. As expected, the sparseness of the measurement set had definite bearing on this quality.

USE OF THE INVENTION

Our invention makes possible a number of core tasks related to the analysis and synthesis of the human figure motion:

Track figure motion in an image sequence using learned dynamic models.

Classify different types of human motion.

Synthesize motion using stochastic models that correspond to different types of motion.

Interpolate missing motion data from sparsely observed image sequences.

We anticipate that our invention could impact the following application areas:

Surveillance: Use of accurate dynamic models could lead to improved tracking in noisy video footage. The ability to interpolate missing data could be useful in situations where frame rates are low, as in Web or other network applications. The ability to classify motion into categories can benefit from the SLDS approach. Two forms of classification are possible. First, specific actions such as “opening a door” or “dropping a package” can be modeled using our approach. Second, it may be possible to recognize specific individuals based on the observed dynamics of their motion in image sequences. This could be used, for example, to recognize criminal suspects using surveillance cameras in public places such as airports or train stations.

User-interfaces: Interfaces based on vision sensing could benefit from improved tracking and better classification performance due to the SLDS approach.

Motion capture: Motion capture in unstructured environments can be enabled through better tracking techniques. In addition to the capture of live motion without the use of special clothing, it is also possible to capture motion from archival sources such as old movies.

Motion synthesis: The generation of human motion for computer graphics animation can be enabled through the use of a learned, generative stochastic model. By learning models from sample motions, it is possible to capture the natural dynamics implicit in real human motion without a laborious manual modeling process. Because the resulting models are stochastic, sampling from the models produces motion with a pleasing degree of randomness.

Video editing: Tracking algorithms based on powerful dynamic models can simplify the task of segmenting video sequences.

Video compression/decompression: The ability to interpolate a video sequence based on a sparse set of samples could provide a new approach to coding and decoding video sequences containing human or other motion. In practice, human motion is common in video sequences. By transmitting key frames detected using SLDS classification at a low sampling rate and interpolating, using SLDS interpolation, the missing frames from the transmitted model parameters, a substantial savings in bit-rate may be achievable.

It will be apparent to those of ordinary skill in the art that methods involved in the present system for learning SLDS models may be embodied in a computer program product that includes a computer usable medium. For example, such a computer usable medium can include a readable memory device, such as a hard drive device, a CD-ROM, a DVD-ROM, or a computer diskette, having computer readable program code segments stored thereon. The computer readable medium can also include a communications or transmission medium, such as a bus or a communications link, either optical, wired, or wireless, having program code segments carried thereon as digital or analog data signals.

While this invention has been particularly shown and described with references to preferred embodiments thereof, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the scope of the invention encompassed by the appended claims. 

What is claimed is:
 1. A method for determining, from a set of possible switching states and responsive to a sequence of measurements, a corresponding sequence of switching states for a system comprising a plurality of dynamic models, the method comprising: associating each model with a switching state, a model being selected when its associated switching state is true; determining a state transition record by determining and recording, for a given measurement and for each possible switching state, an optimal prior switching state, based on the measurement sequence, wherein the optimal prior switching state optimizes a transition probability; determining, for a final measurement, an optimal final switching state; and determining the sequence of switching states by backtracking, from said optimal final switching state, through the state transition record.
 2. The method of claim 1, wherein the switching states comprise a Markov chain.
 3. The method of claim 1, wherein the transition probability is based on likelihood of the measurement sequence and probability of transition between the switching states.
 4. The method of claim 3, wherein the likelihood of the measurement sequence is determined using a Kalman filter.
 5. The method of claim 1, wherein the sequence of measurements is a time sequence, each measurement corresponding to a distinct time.
 6. The method of claim 1, further comprising: learning parameters of the dynamic models, responsive to the determined sequence of switching states resulting from training data.
 7. The method of claim 1, wherein the sequence of measurements comprises economic data.
 8. The method of claim 1, wherein the sequence of measurements comprises image data.
 9. The method of claim 1, wherein the sequence of measurements comprises audio data.
 10. The method of claim 9, wherein audio data comprises speech.
 11. The method of claim 10, wherein the speech comprises phonemes.
 12. The method of claim 1, wherein the sequence of measurements comprises spatial data.
 13. A method for determining, from a set of possible switching states and responsive to a sequence of measurements, a corresponding sequence of switching states for a system comprising a plurality of dynamic models, the method comprising: associating each model with a switching state, a model being selected when its associated switching state is true; and for a given point in the sequence of measurements, for each possible switching state, determining a probability of a transition from each possible prior switching state to said possible switching state, responsive to the measurement for the instant point, and for each possible switching state, selecting the transition having the highest of said determined probabilities, and recording the prior switching state associated with the selected transition.
 14. The method of claim 13, further comprising: repeating, for each of a plurality of sequential points in the sequence of measurements, the steps of determining transition probabilities, selecting a transition and recording the prior state; for the final point in the plurality of sequential points, determining a probability for each possible switching state given the measurement for the final point; and determining a sequence of switching states corresponding to the plurality of sequential measurement points by backtracking, from the final point, through the recorded prior states.
 15. The method of claim 13, wherein the dynamic systems are linear dynamic systems.
 16. A switching linear dynamic system (SLDS) model, comprising: a plurality of linear dynamic system (LDS) models, wherein at any given instance, an LDS model is selected responsive to a switching variable; a state transition recorder which determines a state transition record by determining and recording, for a given measurement and for each possible switching state, an optimal prior switching state, based on a measurement sequence, wherein the optimal prior switching state optimizes a transition probability, and which determines, for a final measurement, an optimal final switching state; and a backtracker which determines a sequence of switching states corresponding to the measurement sequence by backtracking, from said optimal final switching state, through the state transition record.
 17. The SLDS model of claim 16, wherein each LDS model is associated with a switching state, an LDS model being selected when the switching variable holds the associated state.
 18. The SLDS model of claim 16, wherein the switching variable is driven by a hidden markov model (HMM) process.
 19. The SLDS model of claim 16, wherein the transition probability is based on likelihood of the measurement sequence and probability of transition between the switching states.
 20. The SLDS model of claim 19, wherein the likelihood of the measurement sequence is determined using a Kalman filter.
 21. The SLDS model of claim 16, wherein the sequence of measurements is a time sequence, each measurement corresponding to a distinct time.
 22. The SLDS model of claim 16, further comprising: a learning module which learns parameters of the dynamic models responsive to the determined sequence of switching states resulting from training data.
 23. The SLDS model of claim 16, wherein the sequence of measurements comprises economic data.
 24. The SLDS model of claim 16, wherein the sequence of measurements comprises image data.
 25. The SLDS model of claim 16, wherein the sequence of measurements comprises audio data.
 26. The SLDS model of claim 16, wherein the sequence of measurements comprises spatial data.
 27. A switching linear dynamic system (SLDS) model, comprising: a plurality of linear dynamic system (LDS) models, wherein at any given instance, an LDS model is selected responsive to a switching variable; an SLDS model learning parameter module which learns parameters of at least one SLDS model responsive to state space trajectory data; and an approximate Viterbi state sequence inference module, which reestimates parameters of each SLDS model, using Viterbi inference, to minimize a modeling cost of current state sequence estimates.
 28. A system for determining, from a set of possible switching states and responsive to a sequence of measurements, a corresponding sequence of switching states for a system comprising a plurality of dynamic models, the method comprising: means for associating each model with a switching state, a model being selected when its associated switching state is true; means for determining a state transition record by determining and recording, for a given measurement and for each possible switching state, an optimal prior switching state, based on the measurement sequence, wherein the optimal prior switching state optimizes a transition probability; means for determining, for a final measurement, an optimal final switching state; and means for determining the sequence of switching states by backtracking, from said optimal final switching state, through the state transition record.
 29. A computer program product for determining, from a set of possible switching states and responsive to a sequence of measurements, a corresponding sequence of switching states for a system comprising a plurality of dynamic models, the computer program product comprising a computer usable medium having computer readable code thereon, including program code which: associates each model with a switching state, a model being selected when its associated switching state is true; determines a state transition record by determining and recording, for a given measurement and for each possible switching state, an optimal prior switching state, based on the measurement sequence, wherein the optimal prior switching state optimizes a transition probability; determines an optimal final switching state for a final measurement; and determines the sequence of switching states by backtracking, from said optimal final switching state, through the state transition record.
 30. A computer system comprising: a processor; a memory system connected to the processor; and a computer program, in the memory, which: associates each of a plurality of dynamic models with a switching state, a model being selected when its associated switching state is true; determines, from a set of possible switching states and responsive to a sequence of measurements, a state transition record by determining and recording, for a given measurement and for each possible switching state, an optimal prior switching state, wherein the optimal prior switching state optimizes a transition probability; determines an optimal final switching state for a final measurement; and determines a sequence of switching states corresponding to the measurement sequence by backtracking, from said optimal final switching state, through the state transition record.
 31. A computer data signal embodied in a carrier wave for determining, from a set of possible switching states and responsive to a sequence of measurements, a corresponding sequence of switching states for a system comprising a plurality of dynamic models, comprising: program code for associating each model with a switching state, a model being selected when its associated switching state is true; program code for determining a state transition record by determining and recording, for a given measurement and for each possible switching state, an optimal prior switching state, based on the measurement sequence, wherein the optimal prior switching state optimizes a transition probability; program code for determining, for a final measurement, an optimal final switching state; and program code for determining the sequence of switching states by backtracking, from said optimal final switching state, through the state transition record.
 32. A method for determining, from a set of possible switching states and responsive to a sequence of measurements, a corresponding sequence of switching states for a system comprising a plurality of dynamic models, the method comprising: associating each dynamic model with a switching state, a dynamic model being selected when its associated switching state is true, wherein the switching state at a particular instance is determined by a switching model; decoupling the dynamic models from the switching model; determining parameters of a decoupled dynamic model, responsive to a switching state probability estimate; estimating a state of the decoupled dynamic model corresponding to a measurement at the particular instance, and responsive to the sequence of measurements; determining parameters of the decoupled switching model, responsive to the dynamic state estimate; estimating a probability for each possible switching state of the decoupled switching model; and determining the sequence of switching states based on the estimated switching state probabilities.
 33. The method of claim 32, wherein estimating a state of the decoupled dynamic model comprises using a Kalman filter.
 34. The method of claim 32, wherein the decoupled dynamic model is time-varying.
 35. The method of claim 32, further comprising: repeating, until a desired margin of error has been achieved, the steps of determining parameters of the decoupled dynamic model, estimating a dynamic model state, determining parameters of the decoupled switching model, estimating a switching state probability, and determining switching states.
 36. The method of claim 32, wherein the sequence is a time sequence, each measurement corresponding to a distinct time.
 37. The method of claim 32, further comprising: providing training data; learning parameters of the dynamic models, responsive to the determined sequence of switching states resulting from the training data.
 38. The method of claim 37, further comprising: predicting future values in a new sequence based on dynamic models learned from training sequences and the current values.
 39. The method of claim 32, wherein the sequence of measurements comprises economic data.
 40. The method of claim 32, wherein the sequence of measurements comprises image data.
 41. The method of claim 32, wherein the sequence of measurements comprises audio data.
 42. The method of claim 32, wherein the sequence of measurements comprises spatial data.
 43. A switching linear dynamic system (SLDS) model, comprising: a plurality of linear dynamic system (LDS) models, wherein at any given instance, an LDS model is selected responsive to a switching variable; a switching model which determines values of the switching variable; and an approximate variational state sequence inference module, which reestimates parameters of each SLDS model, using variational inference, to minimize a modeling cost of current state sequence estimates.
 44. The SLDS model of claim 43, wherein the approximate variational state sequence inference module decouples the dynamic models from the switching model.
 45. The SLDS model of claim 44, wherein the approximate variational state sequence inference module estimates a state of the decoupled dynamic model using a Kalman filter.
 46. The SLDS model of claim 44, wherein the decoupled dynamic model is time-varying.
 47. The SLDS model of claim 44, wherein the approximate variational state sequence inference module repeatedly, until a desired margin of error has been achieved, determines parameters of the decoupled dynamic model, estimates a dynamic model state, determines parameters of the decoupled switching model, estimates a switching state probability, and determines switching states.
 48. The SLDS model of claim 43, wherein the sequence is a time sequence, each measurement corresponding to a distinct time.
 49. The SLDS model of claim 44, further comprising: an SLDS model learning parameter module which learns parameters of at least one SLDS model responsive to state space trajectory data.
 50. The SLDS model of claim 49, wherein the SLDS model predicts future values in a new sequence based on dynamic models learned from training sequences and the current values.
 51. The SLDS model of claim 44, wherein the sequence of measurements comprises economic data.
 52. The SLDS model of claim 44, wherein the sequence of measurements comprises image data.
 53. The SLDS model of claim 44, wherein the sequence of measurements comprises audio data.
 54. The SLDS model of claim 44, wherein the sequence of measurements comprises spatial data.
 55. A system for determining, from a set of possible switching states and responsive to a sequence of measurements, a corresponding sequence of switching states for a system comprising a plurality of dynamic models, the method comprising: means for associating each dynamic model with a switching state, a dynamic model being selected when its associated switching state is true, wherein the switching state at a particular instance is determined by a switching model; means for decoupling the dynamic models from the switching model; means for determining parameters of a decoupled dynamic model, responsive to a switching state probability estimate; means for estimating a state of the decoupled dynamic model corresponding to a measurement at the particular instance, and responsive to the sequence of measurements; means for determining parameters of the decoupled switching model, responsive to the dynamic state estimate; means for estimating a probability for each possible switching state of the decoupled switching model; and means for determining the sequence of switching states based on the estimated switching state probabilities.
 56. A computer program product for determining, from a set of possible switching states and responsive to a sequence of measurements, a corresponding sequence of switching states for a system comprising a plurality of dynamic models, the computer program product comprising a computer usable medium having computer readable code thereon, including program code which: associates each dynamic model with a switching state, a dynamic model being selected when its associated switching state is true, wherein the switching state at a particular instance is determined by a switching model; decouples the dynamic models from the switching model; determines parameters of a decoupled dynamic model, responsive to a switching state probability estimate; estimates a state of the decoupled dynamic model corresponding to a measurement at the particular instance, and responsive to the sequence of measurements; determines parameters of the decoupled switching model, responsive to the dynamic state estimate; estimates a probability for each possible switching state of the decoupled switching model; and determines the sequence of switching states based on the estimated switching state probabilities.
 57. A computer system comprising: a processor; a memory system connected to the processor; and a computer program, in the memory, which: associates each of a plurality of dynamic models with a switching state, a dynamic model being selected when its associated switching state is true, wherein the switching state at a particular instance is determined by a switching model, the plurality of dynamic models producing a measurement sequence; decouples the dynamic models from the switching model; determines a decoupled dynamic model, responsive to a switching state probability estimate; estimates a state of the decoupled dynamic model corresponding to a measurement at a particular instance, and responsive to the sequence of measurements; determines the decoupled switching model, responsive to the dynamic state estimate; estimates a probability for each possible switching state of the decoupled switching model; and determines a sequence of switching states corresponding to the measurement sequence based on the estimated switching state probabilities.
 58. A computer data signal embodied in a carrier wave for determining, from a set of possible switching states and responsive to a sequence of measurements, a corresponding sequence of switching states for a system comprising a plurality of dynamic models, comprising: program code for associating each dynamic model with a switching state, a dynamic model being selected when its associated switching state is true, wherein the switching state at a particular instance is determined by a switching model; program code for decoupling the dynamic model from the switching model; program code for determining parameters of the decoupled dynamic model, responsive to a switching state probability estimate; program code for estimating a state of the decoupled dynamic model corresponding to a measurement at the particular instance, and responsive to the sequence of measurements; program code for determining parameters of the decoupled switching model, responsive to the dynamic state estimate; program code for estimating a probability for each possible switching state of the decoupled switching model; and program code for determining the sequence of switching states based on the estimated switching state probabilities. 